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id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다" class=""><span>계정 만들기</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o" class=""><span>로그인</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="더 많은 옵션" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="개인 도구" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">개인 도구</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="사용자 메뉴" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko"><span>기부</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>계정 만들기</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="사이트"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="목차" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">처음 위치</div> </a> </li> <li id="toc-정의" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>정의</span> </div> </a> <button aria-controls="toc-정의-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>정의 하위섹션 토글하기</span> </button> <ul id="toc-정의-sublist" class="vector-toc-list"> <li id="toc-도약" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#도약"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>도약</span> </div> </a> <ul id="toc-도약-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-성질" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#성질"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>성질</span> </div> </a> <button aria-controls="toc-성질-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>성질 하위섹션 토글하기</span> </button> <ul id="toc-성질-sublist" class="vector-toc-list"> <li id="toc-함의_관계" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#함의_관계"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>함의 관계</span> </div> </a> <ul id="toc-함의_관계-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-연산" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#연산"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>연산</span> </div> </a> <ul id="toc-연산-sublist" class="vector-toc-list"> <li id="toc-사전식_순서" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#사전식_순서"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>사전식 순서</span> </div> </a> <ul id="toc-사전식_순서-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-위상수학적_성질" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#위상수학적_성질"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>위상수학적 성질</span> </div> </a> <ul id="toc-위상수학적_성질-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-범주론적_성질" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#범주론적_성질"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>범주론적 성질</span> </div> </a> <ul id="toc-범주론적_성질-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-분류" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#분류"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>분류</span> </div> </a> <button aria-controls="toc-분류-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>분류 하위섹션 토글하기</span> </button> <ul id="toc-분류-sublist" class="vector-toc-list"> <li id="toc-가산_조밀_전순서_집합" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#가산_조밀_전순서_집합"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>가산 조밀 전순서 집합</span> </div> </a> <ul id="toc-가산_조밀_전순서_집합-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-완비_분해_가능_조밀_전순서_집합" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#완비_분해_가능_조밀_전순서_집합"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>완비 분해 가능 조밀 전순서 집합</span> </div> </a> <ul id="toc-완비_분해_가능_조밀_전순서_집합-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-유한_집합_위의_(원)전순서" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#유한_집합_위의_(원)전순서"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>유한 집합 위의 (원)전순서</span> </div> </a> <ul id="toc-유한_집합_위의_(원)전순서-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-예" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#예"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>예</span> </div> </a> <button aria-controls="toc-예-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>예 하위섹션 토글하기</span> </button> <ul id="toc-예-sublist" class="vector-toc-list"> <li id="toc-아론샤인_직선" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#아론샤인_직선"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>아론샤인 직선</span> </div> </a> <ul id="toc-아론샤인_직선-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-컨트리먼_직선" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#컨트리먼_직선"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>컨트리먼 직선</span> </div> </a> <ul id="toc-컨트리먼_직선-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-역사" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#역사"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>역사</span> </div> </a> <ul id="toc-역사-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-각주" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#각주"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>각주</span> </div> </a> <ul id="toc-각주-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-외부_링크" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#외부_링크"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>외부 링크</span> </div> </a> <ul id="toc-외부_링크-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="목차" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="목차 토글" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">목차 토글</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">전순서 집합</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="다른 언어로 문서를 방문합니다. 28개 언어로 읽을 수 있습니다" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-28" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">28개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%B1%D8%AA%D9%8A%D8%A8_%D9%83%D9%84%D9%8A" title="ترتيب كلي – 아랍어" lang="ar" hreflang="ar" data-title="ترتيب كلي" data-language-autonym="العربية" data-language-local-name="아랍어" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Ordre_total" title="Ordre total – 카탈로니아어" lang="ca" hreflang="ca" data-title="Ordre total" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs badge-Q70893996 mw-list-item" title=""><a href="https://cs.wikipedia.org/wiki/Line%C3%A1rn%C3%AD_uspo%C5%99%C3%A1d%C3%A1n%C3%AD" title="Lineární uspořádání – 체코어" lang="cs" hreflang="cs" data-title="Lineární uspořádání" data-language-autonym="Čeština" data-language-local-name="체코어" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B8%D0%BB%D0%BB%D0%B5_%D0%B9%C4%95%D1%80%D0%BA%D0%B5%D0%BB%D0%B5%D0%BD%D0%BD%C4%95_%D0%B9%D1%8B%D1%88" title="Линилле йĕркеленнĕ йыш – 추바시어" lang="cv" hreflang="cv" data-title="Линилле йĕркеленнĕ йыш" data-language-autonym="Чӑвашла" data-language-local-name="추바시어" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Total_ordning" title="Total ordning – 덴마크어" lang="da" hreflang="da" data-title="Total ordning" data-language-autonym="Dansk" data-language-local-name="덴마크어" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Totalordnung" title="Totalordnung – 독일어" lang="de" hreflang="de" data-title="Totalordnung" data-language-autonym="Deutsch" data-language-local-name="독일어" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Total_order" title="Total order – 영어" lang="en" hreflang="en" data-title="Total order" data-language-autonym="English" data-language-local-name="영어" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Totala_ordo" title="Totala ordo – 에스페란토어" lang="eo" hreflang="eo" data-title="Totala ordo" data-language-autonym="Esperanto" data-language-local-name="에스페란토어" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Orden_total" title="Orden total – 스페인어" lang="es" hreflang="es" data-title="Orden total" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lineaarne_j%C3%A4rjestus" title="Lineaarne järjestus – 에스토니아어" lang="et" hreflang="et" data-title="Lineaarne järjestus" data-language-autonym="Eesti" data-language-local-name="에스토니아어" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B1%D8%AA%DB%8C%D8%A8_%DA%A9%D9%84%DB%8C" title="ترتیب کلی – 페르시아어" lang="fa" hreflang="fa" data-title="ترتیب کلی" data-language-autonym="فارسی" data-language-local-name="페르시아어" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Ordre_total" title="Ordre total – 프랑스어" lang="fr" hreflang="fr" data-title="Ordre total" data-language-autonym="Français" data-language-local-name="프랑스어" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Orde_total" title="Orde total – 갈리시아어" lang="gl" hreflang="gl" data-title="Orde total" data-language-autonym="Galego" 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class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Total_orden" title="Total orden – 노르웨이어(보크말)" lang="nb" hreflang="nb" data-title="Total orden" data-language-autonym="Norsk bokmål" data-language-local-name="노르웨이어(보크말)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Porz%C4%85dek_liniowy" title="Porządek liniowy – 폴란드어" lang="pl" hreflang="pl" data-title="Porządek liniowy" data-language-autonym="Polski" data-language-local-name="폴란드어" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Rela%C8%9Bie_de_ordine_total%C4%83" title="Relație de ordine totală – 루마니아어" lang="ro" hreflang="ro" data-title="Relație de ordine totală" data-language-autonym="Română" 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class="nowrap"></span> <a href="/wiki/%EC%88%9C%EC%84%9C%EB%A1%A0" title="순서론">순서론</a>에서 <b>전순서 집합</b>(全順序集合, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">totally ordered set, toset</span>)는 임의의 두 <a href="/wiki/%EC%9B%90%EC%86%8C_(%EC%88%98%ED%95%99)" title="원소 (수학)">원소</a>를 비교할 수 있는 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9" title="부분 순서 집합">부분 순서 집합</a>이다. <a href="/wiki/%EC%8B%A4%EC%88%98" title="실수">실수</a>에서는 순서를 줄 수 있지만 <a href="/wiki/%ED%97%88%EC%88%98" title="허수">허수</a>와 <a href="/wiki/%EB%B3%B5%EC%86%8C%EC%88%98" title="복소수">복소수</a>에서는 순서를 줄 수 없다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="정의"><span id=".EC.A0.95.EC.9D.98"></span>정의</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=1" title="부분 편집: 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EC%9B%90%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9" title="원순서 집합">원순서 집합</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\lesssim )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≲</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\lesssim )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0cefd1633a866ad968e394c3b858334f171f0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\lesssim )}"></span>이 다음 조건을 만족시킨다면, <b>원전순서 집합</b>(原全順序集合, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">pretotally ordered set</span>, <span lang="en">totally preordered set</span>, <span lang="en">weakly ordered set</span>)이라고 한다. </p> <ul><li>임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d72f66ab332ed430aa9b34ff18c9723c4fea2a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.34ex; height:2.509ex;" alt="{\displaystyle x,y\in X}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\lesssim y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≲</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\lesssim y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d193e8716d3eda6c6900b2019667fb4c6b6e9e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.584ex; height:2.843ex;" alt="{\displaystyle x\lesssim y}"></span>이거나 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\lesssim x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≲</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\lesssim x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b2154c402edcfc3919a5b337778f8a4765d19c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.584ex; height:2.843ex;" alt="{\displaystyle y\lesssim x}"></span>이다.</li></ul> <p>즉, 두 원소가 항상 비교 가능한 <a href="/wiki/%EC%9B%90%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9" title="원순서 집합">원순서 집합</a>이다. </p><p><b>전순서 집합</b>(全順序集合, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">totally ordered set</span>, <span lang="en">toset</span>)은 원전순서 집합인 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9" title="부분 순서 집합">부분 순서 집합</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span>이다. 즉, <a href="/wiki/%EC%9D%B4%ED%95%AD_%EA%B4%80%EA%B3%84" title="이항 관계">이항 관계</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }"></span>는 다음 세 조건을 만족시킨다. </p> <ul><li>(<a href="/wiki/%EC%B6%94%EC%9D%B4%EC%A0%81_%EA%B4%80%EA%B3%84" title="추이적 관계">추이성</a>) 만약 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y\leq z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y\leq z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a8611fcfaba012b45d149ab8fb94e2ce390b4c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.77ex; height:2.343ex;" alt="{\displaystyle x\leq y\leq z}"></span>라면 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7d407deef131cba48a58513132e69a4db6ca2fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.516ex; height:2.176ex;" alt="{\displaystyle x\leq z}"></span></li> <li>(<a href="/wiki/%EB%B0%98%EB%8C%80%EC%B9%AD_%EA%B4%80%EA%B3%84" class="mw-redirect" title="반대칭 관계">반대칭성</a>) 임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d72f66ab332ed430aa9b34ff18c9723c4fea2a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.34ex; height:2.509ex;" alt="{\displaystyle x,y\in X}"></span>에 대하여, 만약 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span>이며 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\leq x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≤</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\leq x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7de6a6e4f44d9dfcbfaadbdcf388d4b8a6fed109" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle y\leq x}"></span>라면 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/409a91214d63eabe46ec10ff3cbba689ab687366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}"></span></li> <li>(<a href="/wiki/%EC%99%84%EC%A0%84_%EA%B4%80%EA%B3%84" title="완전 관계">완전성</a>) 항상 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≤</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\leq y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07a0bc023490be1c08e6c33a9cdc93bec908224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}"></span>이거나 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\leq x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≤</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\leq x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7de6a6e4f44d9dfcbfaadbdcf388d4b8a6fed109" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle y\leq x}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="도약"><span id=".EB.8F.84.EC.95.BD"></span>도약</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=2" title="부분 편집: 도약"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>전순서 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span>의 <b>도약</b>(跳躍, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">jump</span>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)\in X^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\in X^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab8319f4ba73f5098abfdf8f24d9f32659f248ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.962ex; height:3.176ex;" alt="{\displaystyle (a,b)\in X^{2}}"></span>은 다음 두 조건을 만족시키는 <a href="/wiki/%EC%88%9C%EC%84%9C%EC%8C%8D" title="순서쌍">순서쌍</a>이다. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span>이다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<c<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>c</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<c<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b5555ee657247e2c7c2b930cc75ba2349395d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.431ex; height:2.176ex;" alt="{\displaystyle a<c<b}"></span>인 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20a6fd8987f71d0e8b6f844f05339748989a1267" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.828ex; height:2.176ex;" alt="{\displaystyle c\in X}"></span>가 존재하지 않는다.</li></ul> <p>도약이 없는 전순서를 <a href="/wiki/%EC%A1%B0%EB%B0%80_%EC%88%9C%EC%84%9C" title="조밀 순서">조밀 순서</a>라고 한다. </p> <div class="mw-heading mw-heading2"><h2 id="성질"><span id=".EC.84.B1.EC.A7.88"></span>성질</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=3" title="부분 편집: 성질"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="함의_관계"><span id=".ED.95.A8.EC.9D.98_.EA.B4.80.EA.B3.84"></span>함의 관계</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=4" title="부분 편집: 함의 관계"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>다음과 같은 함의 관계가 성립한다. </p> <dl><dd><table style="text-align: center"> <tbody><tr> <td>전순서 집합</td> <td>⇒</td> <td>원전순서 집합 </td></tr> <tr> <td>⇓</td> <td></td> <td>⇓ </td></tr> <tr> <td><a href="/wiki/%EB%B6%80%EB%B6%84_%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9" title="부분 순서 집합">부분 순서 집합</a></td> <td>⇒</td> <td><a href="/wiki/%EC%9B%90%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9" title="원순서 집합">원순서 집합</a> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="연산"><span id=".EC.97.B0.EC.82.B0"></span>연산</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=5" title="부분 편집: 연산"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EC%9B%90%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9" title="원순서 집합">원순서 집합</a>들의 족 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(X_{i},\lesssim _{i})\}_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mo>≲</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(X_{i},\lesssim _{i})\}_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f0b70e140e3d2893e9055b7ac232fed8cd15c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.224ex; height:2.843ex;" alt="{\displaystyle \{(X_{i},\lesssim _{i})\}_{i\in I}}"></span>가 주어졌으며, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>에 역시 <a href="/wiki/%EC%9B%90%EC%88%9C%EC%84%9C" class="mw-redirect" title="원순서">원순서</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lesssim _{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>≲</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lesssim _{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5235d44bead535a4e427c419a0bbd58c24c4c9b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.869ex; height:2.843ex;" alt="{\displaystyle \lesssim _{I}}"></span>가 부여되었다고 하자. 그렇다면, <a href="/wiki/%EB%B6%84%EB%A6%AC%ED%95%A9%EC%A7%91%ED%95%A9" title="분리합집합">분리합집합</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\textstyle \bigsqcup _{i\in I}X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>⨆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\textstyle \bigsqcup _{i\in I}X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf9d094521b4156e3369c8aa13fefb3386be13c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.85ex; height:3.009ex;" alt="{\displaystyle X=\textstyle \bigsqcup _{i\in I}X_{i}}"></span> 위에 다음과 같은 원순서를 정의할 수 있다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\lesssim y_{j}\iff \left((i\prec _{I}j)\lor \left(i=j\land x_{i}\lesssim _{i}y_{j}\right)\right)\qquad (x_{i}\in X_{i},\;y_{j}\in X_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≲</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <msub> <mo>≺</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mi>j</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>=</mo> <mi>j</mi> <mo>∧</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo>≲</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\lesssim y_{j}\iff \left((i\prec _{I}j)\lor \left(i=j\land x_{i}\lesssim _{i}y_{j}\right)\right)\qquad (x_{i}\in X_{i},\;y_{j}\in X_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7131605fae4430f9ed2b38b8bf9deff1e10ec34e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:67.56ex; height:3.009ex;" alt="{\displaystyle x_{i}\lesssim y_{j}\iff \left((i\prec _{I}j)\lor \left(i=j\land x_{i}\lesssim _{i}y_{j}\right)\right)\qquad (x_{i}\in X_{i},\;y_{j}\in X_{j})}"></span></dd></dl> <p>이를 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(X_{i},\lesssim _{i})\}_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mo>≲</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{(X_{i},\lesssim _{i})\}_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f0b70e140e3d2893e9055b7ac232fed8cd15c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.224ex; height:2.843ex;" alt="{\displaystyle \{(X_{i},\lesssim _{i})\}_{i\in I}}"></span>들의 <b>순서합</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">ordered sum</span>)이라고 한다. (여기서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\prec _{I}j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msub> <mo>≺</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\prec _{I}j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2153a378fc167561cff85824902f474722d2716" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.92ex; height:2.509ex;" alt="{\displaystyle i\prec _{I}j}"></span>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\lesssim _{I}j\not \lesssim _{I}i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msub> <mo>≲</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mi>j</mi> <msub> <mo>≴</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\lesssim _{I}j\not \lesssim _{I}i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac03bad81235f2299807e7d650827fe46740fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.883ex; height:2.843ex;" alt="{\displaystyle i\lesssim _{I}j\not \lesssim _{I}i}"></span>를 뜻하며, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\sim _{I}j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\sim _{I}j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5399fcc3b4ff0e27c78ec17ce2783a54600e56c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.92ex; height:2.509ex;" alt="{\displaystyle i\sim _{I}j}"></span>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\lesssim j\lesssim i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>≲</mo> <mi>j</mi> <mo>≲</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\lesssim j\lesssim i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/488ff16752c7d5b8de5ca0e4ec10929852a3deb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.76ex; height:2.843ex;" alt="{\displaystyle i\lesssim j\lesssim i}"></span>를 뜻한다.) </p><p>이에 대하여 다음이 성립한다. </p> <ul><li>만약 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (I,\lesssim _{I})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>I</mi> <mo>,</mo> <msub> <mo>≲</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (I,\lesssim _{I})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a96aeb6bf43f1f216065de44111f57fd99db6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle (I,\lesssim _{I})}"></span>가 전순서 집합이며, 모든 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{i},\lesssim _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mo>≲</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{i},\lesssim _{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fba9ef7703a4aae0a76e8479584106f33948bd5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.175ex; height:2.843ex;" alt="{\displaystyle (X_{i},\lesssim _{i})}"></span>가 원전순서 집합이라면, 그 순서합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> 역시 원전순서 집합이다.</li> <li>만약 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (I,\lesssim _{I})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>I</mi> <mo>,</mo> <msub> <mo>≲</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (I,\lesssim _{I})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a96aeb6bf43f1f216065de44111f57fd99db6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle (I,\lesssim _{I})}"></span>가 전순서 집합이며, 모든 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{i},\lesssim _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mo>≲</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{i},\lesssim _{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fba9ef7703a4aae0a76e8479584106f33948bd5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.175ex; height:2.843ex;" alt="{\displaystyle (X_{i},\lesssim _{i})}"></span>가 전순서 집합이라면, 그 순서합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> 역시 전순서 집합이다.</li> <li>만약 모든 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{i},\lesssim _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mo>≲</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{i},\lesssim _{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fba9ef7703a4aae0a76e8479584106f33948bd5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.175ex; height:2.843ex;" alt="{\displaystyle (X_{i},\lesssim _{i})}"></span>가 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9" title="부분 순서 집합">부분 순서 집합</a>이라면, 그 순서합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> 역시 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9" title="부분 순서 집합">부분 순서 집합</a>이다.</li></ul> <div class="mw-heading mw-heading4"><h4 id="사전식_순서"><span id=".EC.82.AC.EC.A0.84.EC.8B.9D_.EC.88.9C.EC.84.9C"></span>사전식 순서</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=6" title="부분 편집: 사전식 순서"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r34311305">.mw-parser-output .hatnote{}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EC%82%AC%EC%A0%84%EC%8B%9D_%EC%88%9C%EC%84%9C" title="사전식 순서">사전식 순서</a>입니다.</div> <p><a class="mw-selflink selflink">전순서 집합</a>들의 족 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{i},\leq _{i})_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{i},\leq _{i})_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b661ab66c1a427cbe7edcb649b7c6cd57170e6a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.899ex; height:2.843ex;" alt="{\displaystyle (X_{i},\leq _{i})_{i\in I}}"></span>이 주어졌으며, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> 위에 <a href="/wiki/%EC%A0%95%EB%A0%AC_%EC%88%9C%EC%84%9C" class="mw-redirect" title="정렬 순서">정렬 순서</a>가 주어졌을 때, 곱집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{i\in I}X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{i\in I}X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e4cf2ee0af61dd971a31005ff6d10fb3fe3267f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:6.08ex; height:5.676ex;" alt="{\displaystyle \prod _{i\in I}X_{i}}"></span> 위에 <b><a href="/wiki/%EC%82%AC%EC%A0%84%EC%8B%9D_%EC%88%9C%EC%84%9C" title="사전식 순서">사전식 순서</a></b>라는 전순서를 부여할 수 있다. </p> <div class="mw-heading mw-heading3"><h3 id="위상수학적_성질"><span id=".EC.9C.84.EC.83.81.EC.88.98.ED.95.99.EC.A0.81_.EC.84.B1.EC.A7.88"></span>위상수학적 성질</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=7" title="부분 편집: 위상수학적 성질"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EC%88%9C%EC%84%9C_%EC%9C%84%EC%83%81" title="순서 위상">순서 위상</a>입니다.</div> <p>원전순서 집합에는 <a href="/wiki/%EC%88%9C%EC%84%9C_%EC%9C%84%EC%83%81" title="순서 위상">순서 위상</a>을 부여하여 <a href="/wiki/%EC%9C%84%EC%83%81_%EA%B3%B5%EA%B0%84_(%EC%88%98%ED%95%99)" title="위상 공간 (수학)">위상 공간</a>으로 취급할 수 있다. </p><p>모든 원전순서 집합은 (<a href="/wiki/%EC%88%9C%EC%84%9C_%EC%9C%84%EC%83%81" title="순서 위상">순서 위상</a> 아래) <a href="/wiki/%EC%99%84%EB%B9%84_%EC%A0%95%EA%B7%9C_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="완비 정규 공간">완비 정규 공간</a>이며, 모든 전순서 집합은 <a href="/wiki/%ED%95%98%EC%9A%B0%EC%8A%A4%EB%8F%84%EB%A5%B4%ED%94%84_%EA%B3%B5%EA%B0%84" title="하우스도르프 공간">하우스도르프</a> <a href="/wiki/%EC%99%84%EB%B9%84_%EC%A0%95%EA%B7%9C_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="완비 정규 공간">완비 정규 공간</a>이다. </p><p>전순서 집합에 대하여, 다음 두 조건이 서로 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>이다. </p> <ul><li><a href="/wiki/%EC%84%A0%ED%98%95_%EC%97%B0%EC%86%8D%EC%B2%B4" title="선형 연속체">선형 연속체</a>이다.</li> <li><a href="/wiki/%EC%88%9C%EC%84%9C_%EC%9C%84%EC%83%81" title="순서 위상">순서 위상</a>을 가했을 때, <a href="/wiki/%EC%97%B0%EA%B2%B0_%EA%B3%B5%EA%B0%84" title="연결 공간">연결 공간</a>이다.</li></ul> <p><a href="/wiki/%EC%99%84%EB%B9%84_%EA%B2%A9%EC%9E%90" title="완비 격자">완비</a> 전순서 집합은 항상 <a href="/wiki/%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="콤팩트 공간">콤팩트 공간</a>이다. </p><p>전순서 집합의 부분 공간은 항상 <a href="/wiki/%EC%A7%81%EA%B5%90_%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="직교 콤팩트 공간">직교 콤팩트 공간</a>이자 <a href="/wiki/%EA%B0%80%EC%82%B0_%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="가산 파라콤팩트 공간">가산 파라콤팩트 공간</a>이다. 전순서 집합이 <a href="/wiki/%EB%A9%94%ED%83%80%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="메타콤팩트 공간">메타콤팩트 공간</a>이라면, <a href="/wiki/%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="파라콤팩트 공간">파라콤팩트 공간</a>이다. </p><p>모든 <a href="/wiki/%EB%B6%84%ED%95%B4_%EA%B0%80%EB%8A%A5_%EA%B3%B5%EA%B0%84" title="분해 가능 공간">분해 가능</a> 전순서 집합은 항상 <a href="/wiki/%EC%82%AC%EC%A0%84%EC%8B%9D_%EC%88%9C%EC%84%9C" title="사전식 순서">사전식 순서</a>를 준 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbce239776e28449532ac25f3ac4a003b5cce155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.681ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} \times 2}"></span>의 부분 집합과 순서 동형이다.<sup id="cite_ref-Geschke_1-0" class="reference"><a href="#cite_note-Geschke-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:Theorem 4</sup></span> </p> <div class="mw-heading mw-heading3"><h3 id="범주론적_성질"><span id=".EB.B2.94.EC.A3.BC.EB.A1.A0.EC.A0.81_.EC.84.B1.EC.A7.88"></span>범주론적 성질</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=8" title="부분 편집: 범주론적 성질"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>전순서 집합과 <a href="/wiki/%EC%A6%9D%EA%B0%80_%ED%95%A8%EC%88%98" class="mw-redirect" title="증가 함수">증가 함수</a>는 <a href="/wiki/%EA%B5%AC%EC%B2%B4%EC%A0%81_%EB%B2%94%EC%A3%BC" title="구체적 범주">구체적 범주</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Toset} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Toset</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Toset} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2c131f6a202f6a4047fb7d73d74133385bf53c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.694ex; height:2.176ex;" alt="{\displaystyle \operatorname {Toset} }"></span>를 이룬다. 이는 <a href="/wiki/%EC%9E%91%EC%9D%80_%EB%B2%94%EC%A3%BC" title="작은 범주">작은 범주</a>의 범주 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cat} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Cat</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Cat} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/427df30925c7b90781486f30861faf68e076643f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.745ex; height:2.176ex;" alt="{\displaystyle \operatorname {Cat} }"></span>의 <a href="/wiki/%EC%B6%A9%EB%A7%8C%ED%95%9C_%EB%B6%80%EB%B6%84_%EB%B2%94%EC%A3%BC" class="mw-redirect" title="충만한 부분 범주">충만한 부분 범주</a>이다. </p><p>공집합이 아닌 유한 전순서 집합들의 범주 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d909fe94e8277a4c44a50853cb7dbbf0fa3148ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle \triangle }"></span>는 <b>단체 범주</b>(單體範疇, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">simplex category</span>)라고 하며, 그 위의 <a href="/wiki/%EC%A4%80%EC%B8%B5" title="준층">준층</a> 범주 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {PSh} (\triangle )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>PSh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">△</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {PSh} (\triangle )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6bafe3194197e011935dc86a4ae909f4714612" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.043ex; height:2.843ex;" alt="{\displaystyle \operatorname {PSh} (\triangle )}"></span>는 <b><a href="/wiki/%EB%8B%A8%EC%B2%B4_%EC%A7%91%ED%95%A9" title="단체 집합">단체 집합</a></b>이라고 한다. 이는 <a href="/wiki/%ED%98%B8%EB%AA%A8%ED%86%A0%ED%94%BC_%EC%9D%B4%EB%A1%A0" class="mw-redirect" title="호모토피 이론">호모토피 이론</a>에서 매우 중요하게 사용된다. </p> <div class="mw-heading mw-heading2"><h2 id="분류"><span id=".EB.B6.84.EB.A5.98"></span>분류</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=9" title="부분 편집: 분류"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>모든 전순서 집합의 분류는 <a href="/wiki/%EC%84%A0%ED%83%9D_%EA%B3%B5%EB%A6%AC" title="선택 공리">선택 공리</a>를 추가한 <a href="/wiki/%EC%B2%B4%EB%A5%B4%EB%A9%9C%EB%A1%9C-%ED%94%84%EB%A0%9D%EC%BC%88_%EC%A7%91%ED%95%A9%EB%A1%A0" title="체르멜로-프렝켈 집합론">체르멜로-프렝켈 집합론</a> 속에서는 불가능하다. 예를 들어, 비교적 간단한 분류 문제인 <a href="/wiki/%EC%88%98%EC%8A%AC%EB%A6%B0_%EA%B0%80%EC%84%A4" class="mw-redirect" title="수슬린 가설">수슬린 가설</a>조차 증명하거나 반증할 수 없다. 그러나 특수한 경우에는 다음과 같은 분류 정리가 존재한다. </p> <div class="mw-heading mw-heading3"><h3 id="가산_조밀_전순서_집합"><span id=".EA.B0.80.EC.82.B0_.EC.A1.B0.EB.B0.80_.EC.A0.84.EC.88.9C.EC.84.9C_.EC.A7.91.ED.95.A9"></span>가산 조밀 전순서 집합</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=10" title="부분 편집: 가산 조밀 전순서 집합"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EC%A1%B0%EB%B0%80_%EC%88%9C%EC%84%9C" title="조밀 순서">조밀</a> <a href="/wiki/%EA%B0%80%EC%82%B0_%EC%A7%91%ED%95%A9" title="가산 집합">가산</a> 전순서 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span>은 다음 여섯 <a class="mw-selflink selflink">전순서 집합</a> 가운데 정확히 하나와 순서 동형이다. </p> <ul><li><a href="/wiki/%EA%B3%B5%EC%A7%91%ED%95%A9" title="공집합">공집합</a></li> <li><a href="/wiki/%ED%95%9C%EC%9B%90%EC%86%8C_%EC%A7%91%ED%95%A9" title="한원소 집합">한원소 집합</a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> (<a href="/wiki/%EC%9C%A0%EB%A6%AC%EC%88%98" title="유리수">유리수</a>의 전순서 집합)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{-\infty \}\sqcup \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> <mo>⊔</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{-\infty \}\sqcup \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ad2587cdfb8a8c190cafce2f7d0b83ea80f00f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.848ex; height:2.843ex;" alt="{\displaystyle \{-\infty \}\sqcup \mathbb {Q} }"></span>. 이는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c592030f6141cd6b6f9c887b69ef2a906f2f80bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.141ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{\geq 0}}"></span>과 순서 동형이다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} \sqcup \{+\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>⊔</mo> <mo fence="false" stretchy="false">{</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} \sqcup \{+\infty \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d043c6a7d245836da25ea50f8f7aba23d27e397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.848ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} \sqcup \{+\infty \}}"></span>. 이는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{\leq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≤</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{\leq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6db4de4cd449fcb822c50404b489560e952d4f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.141ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{\leq 0}}"></span>과 순서 동형이다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} \sqcup \{-\infty ,+\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>⊔</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} \sqcup \{-\infty ,+\infty \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b75730edea82bb8f10559ba48a6d110e9fae19f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.013ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} \sqcup \{-\infty ,+\infty \}}"></span>. 이는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} \cap [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>∩</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} \cap [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/822d49c61001a9fd0fee4578855b367df40dc4ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.043ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} \cap [0,1]}"></span>과 순서 동형이다.</li></ul> <p>특히, <a href="/wiki/%EC%B5%9C%EB%8C%80_%EC%9B%90%EC%86%8C" class="mw-redirect" title="최대 원소">최대 원소</a>와 <a href="/wiki/%EC%B5%9C%EC%86%8C_%EC%9B%90%EC%86%8C" class="mw-redirect" title="최소 원소">최소 원소</a>를 갖지 않는 <a href="/wiki/%EB%B6%84%ED%95%B4_%EA%B0%80%EB%8A%A5_%EA%B3%B5%EA%B0%84" title="분해 가능 공간">분해 가능</a> <a href="/wiki/%EC%A1%B0%EB%B0%80_%EC%88%9C%EC%84%9C" title="조밀 순서">조밀</a> <a href="/wiki/%EA%B0%80%EC%82%B0_%EC%A7%91%ED%95%A9" title="가산 집합">가산</a> 전순서 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span>은 항상 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>와 순서 동형이다. </p> <div class="mw-heading mw-heading3"><h3 id="완비_분해_가능_조밀_전순서_집합"><span id=".EC.99.84.EB.B9.84_.EB.B6.84.ED.95.B4_.EA.B0.80.EB.8A.A5_.EC.A1.B0.EB.B0.80_.EC.A0.84.EC.88.9C.EC.84.9C_.EC.A7.91.ED.95.A9"></span>완비 분해 가능 조밀 전순서 집합</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=11" title="부분 편집: 완비 분해 가능 조밀 전순서 집합"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>전순서 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span>가 다음 세 조건을 만족시킨다고 하자. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>는 <a href="/wiki/%EC%A1%B0%EB%B0%80_%EC%88%9C%EC%84%9C" title="조밀 순서">조밀 순서</a>이다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>에 <a href="/wiki/%EC%88%9C%EC%84%9C_%EC%9C%84%EC%83%81" title="순서 위상">순서 위상</a>을 가하면, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>는 <a href="/wiki/%EB%B6%84%ED%95%B4_%EA%B0%80%EB%8A%A5_%EA%B3%B5%EA%B0%84" title="분해 가능 공간">분해 가능 공간</a>이다.</li> <li>(완비성) <a href="/wiki/%EC%83%81%EA%B3%84_(%EC%88%98%ED%95%99)" class="mw-redirect" title="상계 (수학)">상계</a>와 <a href="/wiki/%ED%95%98%EA%B3%84_(%EC%88%98%ED%95%99)" class="mw-redirect" title="하계 (수학)">하계</a>를 갖는 임의의 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%A7%91%ED%95%A9" class="mw-redirect" title="부분 집합">부분 집합</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dce86da0107830a9a97287f9486d9b4ff022875" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.822ex; height:2.343ex;" alt="{\displaystyle A\subseteq X}"></span>는 (만약 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\neq \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>≠</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\neq \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c84bfaec7b2091189e67d3e979e4474a35640e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.65ex; height:2.676ex;" alt="{\displaystyle A\neq \varnothing }"></span>이라면) <a href="/wiki/%EC%83%81%ED%95%9C" class="mw-redirect" title="상한">상한</a>과 <a href="/wiki/%ED%95%98%ED%95%9C" class="mw-redirect" title="하한">하한</a>을 갖는다.</li></ul> <p>그렇다면, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>는 다음 여섯 전순서 집합 가운데 정확히 하나와 순서 동형이다. </p> <ul><li><a href="/wiki/%EA%B3%B5%EC%A7%91%ED%95%A9" title="공집합">공집합</a></li> <li><a href="/wiki/%ED%95%9C%EC%9B%90%EC%86%8C_%EC%A7%91%ED%95%A9" title="한원소 집합">한원소 집합</a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> (<a href="/wiki/%EC%8B%A4%EC%88%98" title="실수">실수</a>의 전순서 집합). 이는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span>과 순서 동형이다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \sqcup \{+\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>⊔</mo> <mo fence="false" stretchy="false">{</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \sqcup \{+\infty \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fcc7a365c66f3253aa9c7da7fccc35022e5334c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.717ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \sqcup \{+\infty \}}"></span>. 이는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e70f9c241f9faa8e9fdda2e8b238e288807d7a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.91ex; height:2.843ex;" alt="{\displaystyle (0,1]}"></span>과 순서 동형이다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \sqcup \{-\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>⊔</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \sqcup \{-\infty \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca322871d95e8eae64844a9a72edc3b1c12951a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.717ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \sqcup \{-\infty \}}"></span>. 이는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f99b30b4451167959e97802252ad13b87af5505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.91ex; height:2.843ex;" alt="{\displaystyle [0,1)}"></span>과 순서 동형이다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\mathbb {R} }}=\mathbb {R} \sqcup \{+\infty ,-\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>⊔</mo> <mo fence="false" stretchy="false">{</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbb {R} }}=\mathbb {R} \sqcup \{+\infty ,-\infty \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa2ce9c84c045ccd74004006ee06f6d89166c95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.66ex; height:3.009ex;" alt="{\displaystyle {\bar {\mathbb {R} }}=\mathbb {R} \sqcup \{+\infty ,-\infty \}}"></span> (<a href="/wiki/%ED%99%95%EC%9E%A5%EB%90%9C_%EC%8B%A4%EC%88%98" title="확장된 실수">확장된 실수</a>). 이는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"></span>과 순서 동형이다.</li></ul> <p>특히, <a href="/wiki/%EC%99%84%EB%B9%84_%EA%B2%A9%EC%9E%90" title="완비 격자">완비</a> <a href="/wiki/%EB%B6%84%ED%95%B4_%EA%B0%80%EB%8A%A5_%EA%B3%B5%EA%B0%84" title="분해 가능 공간">분해 가능</a> <a href="/wiki/%EC%A1%B0%EB%B0%80_%EC%88%9C%EC%84%9C" title="조밀 순서">조밀</a> <a href="/wiki/%EB%AC%B4%ED%95%9C_%EC%A7%91%ED%95%A9" title="무한 집합">무한</a> 전순서 집합은 (순서 동형 아래) <a href="/wiki/%ED%99%95%EC%9E%A5%EB%90%9C_%EC%8B%A4%EC%88%98" title="확장된 실수">확장된 실수</a>의 전순서 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\overline {\mathbb {R} }},\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\overline {\mathbb {R} }},\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4160133e08d2d4e28bb3fb9a5ffb7c9d2b338e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.444ex; height:3.509ex;" alt="{\displaystyle ({\overline {\mathbb {R} }},\leq )}"></span> 밖에 없다. </p> <div class="mw-collapsible mw-collapsed toccolours"> <p><b>증명:</b> </p> <div class="mw-collapsible-content"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>가 위 성질들을 만족시킨다고 하자. <a href="/wiki/%EB%B6%84%ED%95%B4_%EA%B0%80%EB%8A%A5_%EA%B3%B5%EA%B0%84" title="분해 가능 공간">분해 가능 공간</a>의 정의에 의하여, <a href="/wiki/%EA%B0%80%EC%82%B0_%EC%A7%91%ED%95%A9" title="가산 집합">가산</a> <a href="/wiki/%EC%A1%B0%EB%B0%80_%EC%A7%91%ED%95%A9" title="조밀 집합">조밀 집합</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41d3949fca38567b9801a052056fbba32fd0ea11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.003ex; height:2.343ex;" alt="{\displaystyle D\subseteq X}"></span>을 찾을 수 있으며, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> 위의 순서는 <a href="/wiki/%EC%A1%B0%EB%B0%80_%EC%88%9C%EC%84%9C" title="조밀 순서">조밀 순서</a>임을 쉽게 알 수 있다. 따라서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>는 위와 같이 6개의 순서형 가운데 하나와 동형이며, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>의 데데킨트 완비화와 순서 동형이다. </p> </div></div> <p>마지막 조건을 약화시킬 경우, 이들의 분류는 <a href="/wiki/%EC%88%98%EC%8A%AC%EB%A6%B0_%EA%B0%80%EC%84%A4" class="mw-redirect" title="수슬린 가설">수슬린 가설</a>에 의하여 좌우되는데, 이는 <a href="/wiki/%EC%84%A0%ED%83%9D_%EA%B3%B5%EB%A6%AC" title="선택 공리">선택 공리</a>를 추가한 <a href="/wiki/%EC%B2%B4%EB%A5%B4%EB%A9%9C%EB%A1%9C-%ED%94%84%EB%A0%9D%EC%BC%88_%EC%A7%91%ED%95%A9%EB%A1%A0" title="체르멜로-프렝켈 집합론">체르멜로-프렝켈 집합론</a>과 독립적인 명제이다. </p><p>어떤 전순서 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span>에 대하여, 다음 네 조건이 동치이다.<sup id="cite_ref-Geschke_1-1" class="reference"><a href="#cite_note-Geschke-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:Theorem 6</sup></span> </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>는 <a href="/wiki/%EB%B6%84%ED%95%B4_%EA%B0%80%EB%8A%A5_%EA%B3%B5%EA%B0%84" title="분해 가능 공간">분해 가능 공간</a>이며, 가산 개의 도약을 갖는다.</li> <li>다음 조건을 만족시키는 <a href="/wiki/%EA%B0%80%EC%82%B0_%EC%A7%91%ED%95%A9" title="가산 집합">가산 집합</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41d3949fca38567b9801a052056fbba32fd0ea11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.003ex; height:2.343ex;" alt="{\displaystyle D\subseteq X}"></span>가 존재한다. <ul><li>임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\sup\{d\in D\colon d\leq x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <mi>d</mi> <mo>∈</mo> <mi>D</mi> <mo>:</mo> <mi>d</mi> <mo>≤</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\sup\{d\in D\colon d\leq x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5200fb8d085def80cd3c0c393914b6ffc309257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.913ex; height:2.843ex;" alt="{\displaystyle x=\sup\{d\in D\colon d\leq x\}}"></span></li></ul></li> <li>다음 조건을 만족시키는 <a href="/wiki/%EA%B0%80%EC%82%B0_%EC%A7%91%ED%95%A9" title="가산 집합">가산 집합</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41d3949fca38567b9801a052056fbba32fd0ea11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.003ex; height:2.343ex;" alt="{\displaystyle D\subseteq X}"></span>가 존재한다. <ul><li>임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\inf\{d\in D\colon d\geq x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>d</mi> <mo>∈</mo> <mi>D</mi> <mo>:</mo> <mi>d</mi> <mo>≥</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\inf\{d\in D\colon d\geq x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/174a1af0860a7e333d68482af29195e6f0de44f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.063ex; height:2.843ex;" alt="{\displaystyle x=\inf\{d\in D\colon d\geq x\}}"></span></li></ul></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>의 <a href="/wiki/%EB%B6%80%EB%B6%84_%EC%A7%91%ED%95%A9" class="mw-redirect" title="부분 집합">부분 집합</a>과 순서 동형이다. 즉, <a href="/wiki/%EB%8B%A8%EC%82%AC_%ED%95%A8%EC%88%98" title="단사 함수">단사</a> <a href="/wiki/%EB%8B%A8%EC%A1%B0_%ED%95%A8%EC%88%98" class="mw-redirect" title="단조 함수">단조 함수</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359ea801448b482438cb2149cfce6559dc3385b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.585ex; height:2.509ex;" alt="{\displaystyle f\colon X\to \mathbb {R} }"></span>가 존재한다.</li></ul> <div class="mw-heading mw-heading3"><h3 id="유한_집합_위의_(원)전순서"><span id=".EC.9C.A0.ED.95.9C_.EC.A7.91.ED.95.A9_.EC.9C.84.EC.9D.98_.28.EC.9B.90.29.EC.A0.84.EC.88.9C.EC.84.9C"></span>유한 집합 위의 (원)전순서</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=12" title="부분 편집: 유한 집합 위의 (원)전순서"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:13-Weak-Orders.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/13-Weak-Orders.svg/220px-13-Weak-Orders.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/13-Weak-Orders.svg/330px-13-Weak-Orders.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/13-Weak-Orders.svg/440px-13-Weak-Orders.svg.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>크기 3의 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a,b,c\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a,b,c\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75e9bc621ced3f02e87b1c40be37867929142bf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.627ex; height:2.843ex;" alt="{\displaystyle \{a,b,c\}}"></span> 위에 존재할 수 있는 13개의 원전순서. 여기서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\lesssim b\not \lesssim a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≲</mo> <mi>b</mi> <mo>≴</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\lesssim b\not \lesssim a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32750cc75b41d5b289eedc865ddec514d043807f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.655ex; height:2.843ex;" alt="{\displaystyle a\lesssim b\not \lesssim a}"></span>를 뜻한다. 이 가운데 맨 밖의, 검은 색 글씨의 6개는 전순서이다. 중간의, 푸른 색 글씨의 6개는 2개의 <a href="/wiki/%EB%8F%99%EC%B9%98%EB%A5%98" class="mw-redirect" title="동치류">동치류</a>들을 갖는 원전순서이다. 가운데의, 붉은 색 글씨의 1개는 1개의 동치류를 갖는 비이산 원순서이다.</figcaption></figure> <p>유한 전순서 집합은 항상 <a href="/wiki/%EC%A0%95%EB%A0%AC_%EC%A7%91%ED%95%A9" class="mw-redirect" title="정렬 집합">정렬 집합</a>이며, 따라서 그 크기에 따라 완전히 분류된다. </p><p>크기 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>의 <a href="/wiki/%EC%9C%A0%ED%95%9C_%EC%A7%91%ED%95%A9" title="유한 집합">유한 집합</a> 위의 원전순서들의 수는 <b>푸비니 수</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Fubini number</span>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span>이라고 한다.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:228</sup></span> 크기 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>의 유한 집합 위의 전순서들의 수는 <a href="/wiki/%EA%B3%84%EC%8A%B9_(%EC%88%98%ED%95%99)" title="계승 (수학)">계승</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"></span>이다. 이들의 값은 다음과 같다. ((<a href="/wiki/%EC%98%A8%EB%9D%BC%EC%9D%B8_%EC%A0%95%EC%88%98%EC%97%B4_%EC%82%AC%EC%A0%84" title="온라인 정수열 사전">OEIS</a>의 수열 <a href="//oeis.org/A670" class="extiw" title="oeis:A670">A670</a>), (<a href="/wiki/%EC%98%A8%EB%9D%BC%EC%9D%B8_%EC%A0%95%EC%88%98%EC%97%B4_%EC%82%AC%EC%A0%84" title="온라인 정수열 사전">OEIS</a>의 수열 <a href="//oeis.org/A142" class="extiw" title="oeis:A142">A142</a>)). </p> <table class="wikitable" style="text-align: right"> <tbody><tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> </th> <td>0</td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7 </td></tr> <tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span> </th> <td>1</td> <td>1</td> <td>3</td> <td>13</td> <td>75</td> <td>541</td> <td>4683</td> <td>47293 </td></tr> <tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"></span> </th> <td>1</td> <td>1</td> <td>2</td> <td>6</td> <td>24</td> <td>120</td> <td>720</td> <td>5040 </td></tr></tbody></table> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="예"><span id=".EC.98.88"></span>예</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=13" title="부분 편집: 예"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>모든 <a href="/wiki/%EC%88%9C%EC%84%9C%EC%B2%B4" title="순서체">순서체</a>는 전순서 집합이다. 예를 들어, <a href="/wiki/%EC%8B%A4%EC%88%98%EC%B2%B4" class="mw-redirect" title="실수체">실수체</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>, <a href="/wiki/%EC%9C%A0%EB%A6%AC%EC%88%98%EC%B2%B4" class="mw-redirect" title="유리수체">유리수체</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> 등은 표준적인 순서를 부여하면 전순서 집합을 이룬다. <a href="/wiki/%EC%A0%95%EC%88%98%ED%99%98" class="mw-redirect" title="정수환">정수환</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>나 자연수의 <a href="/wiki/%EB%AA%A8%EB%85%B8%EC%9D%B4%EB%93%9C" title="모노이드">모노이드</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span> 역시 전순서 집합이다. 이들 집합 가운데, <a href="/wiki/%EC%9E%90%EC%97%B0%EC%88%98" title="자연수">자연수</a>의 집합을 제외한 나머지는 <a href="/wiki/%EC%A0%95%EB%A0%AC_%EC%A7%91%ED%95%A9" class="mw-redirect" title="정렬 집합">정렬 집합</a>이 아니다. </p> <div class="mw-heading mw-heading3"><h3 id="아론샤인_직선"><span id=".EC.95.84.EB.A1.A0.EC.83.A4.EC.9D.B8_.EC.A7.81.EC.84.A0"></span>아론샤인 직선</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=14" title="부분 편집: 아론샤인 직선"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>아론샤인 직선</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Aronszajn line</span>)은 다음 조건들을 만족시키는 전순서 집합이다.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:43–44, Chapter 14</sup></span> </p> <ul><li><a href="/wiki/%EC%A7%91%ED%95%A9%EC%9D%98_%ED%81%AC%EA%B8%B0" title="집합의 크기">크기</a>가 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}"></span>이다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}"></span> (최소 비가산 <a href="/wiki/%EC%88%9C%EC%84%9C%EC%88%98" title="순서수">순서수</a>)과 순서 동형인 부분 집합을 갖지 않는다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}^{\operatorname {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>op</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}^{\operatorname {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e491cc8073555a8c31b2cf14c47afcb1ae8a3ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.414ex; height:3.176ex;" alt="{\displaystyle \omega _{1}^{\operatorname {op} }}"></span> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}"></span>의 반대 순서)와 순서 동형인 부분 집합을 갖지 않는다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>의 <a href="/wiki/%EB%B9%84%EA%B0%80%EC%82%B0" class="mw-redirect" title="비가산">비가산</a> 부분 집합과 순서 동형인 부분 집합을 갖지 않는다.</li></ul> <p>아론샤인 직선의 존재는 <a href="/wiki/%EC%84%A0%ED%83%9D_%EA%B3%B5%EB%A6%AC" title="선택 공리">선택 공리</a>를 추가한 <a href="/wiki/%EC%B2%B4%EB%A5%B4%EB%A9%9C%EB%A1%9C-%ED%94%84%EB%A0%9D%EC%BC%88_%EC%A7%91%ED%95%A9%EB%A1%A0" title="체르멜로-프렝켈 집합론">체르멜로-프렝켈 집합론</a>만으로 보일 수 있다. 아론샤인 직선은 나흐만 아론샤인(<span style="font-size: smaller;"><a href="/wiki/%ED%8F%B4%EB%9E%80%EB%93%9C%EC%96%B4" title="폴란드어">폴란드어</a>: </span><span lang="pl">Nachman Aronszajn</span>, 1907~1980)이 도입하였다. </p> <div class="mw-heading mw-heading3"><h3 id="컨트리먼_직선"><span id=".EC.BB.A8.ED.8A.B8.EB.A6.AC.EB.A8.BC_.EC.A7.81.EC.84.A0"></span>컨트리먼 직선</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=15" title="부분 편집: 컨트리먼 직선"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>원순서 집합들의 족 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\lesssim )_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≲</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\lesssim )_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c7fb2d73de88200fa5e925c524eb7acb2801a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.356ex; height:2.843ex;" alt="{\displaystyle (X,\lesssim )_{i\in I}}"></span>이 주어졌을 때, <a href="/wiki/%EA%B3%B1%EC%A7%91%ED%95%A9" title="곱집합">곱집합</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \prod _{i\in I}X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \prod _{i\in I}X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82078cf3815a331cf25af4bb7d969cdb46bd8f6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.029ex; height:3.009ex;" alt="{\displaystyle \textstyle \prod _{i\in I}X_{i}}"></span> 위에 <a href="/wiki/%EC%9B%90%EC%88%9C%EC%84%9C" class="mw-redirect" title="원순서">원순서</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\lesssim y\iff \forall i\in I\colon x_{i}\lesssim y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≲</mo> <mi>y</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≲</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\lesssim y\iff \forall i\in I\colon x_{i}\lesssim y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b98a6d84dc0b7a9a31fd8dbb21bc9c559315e019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.789ex; height:2.843ex;" alt="{\displaystyle x\lesssim y\iff \forall i\in I\colon x_{i}\lesssim y_{i}}"></span></dd></dl> <p>를 줄 수 있다. 마찬가지로, <a href="/wiki/%EB%B6%84%EB%A6%AC%ED%95%A9%EC%A7%91%ED%95%A9" title="분리합집합">분리합집합</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \bigsqcup _{i\in I}X_{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>⨆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \bigsqcup _{i\in I}X_{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a95c35400eb896fc52e76166415db270a75f5f1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.033ex; height:3.009ex;" alt="{\displaystyle \textstyle \bigsqcup _{i\in I}X_{I}}"></span> 위에 <a href="/wiki/%EC%9B%90%EC%88%9C%EC%84%9C" class="mw-redirect" title="원순서">원순서</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\lesssim y\iff \exists i\in I\colon x\in X_{i}\ni y\land x\lesssim _{i}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≲</mo> <mi>y</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∋</mo> <mi>y</mi> <mo>∧</mo> <mi>x</mi> <msub> <mo>≲</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\lesssim y\iff \exists i\in I\colon x\in X_{i}\ni y\land x\lesssim _{i}y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb36346fd4a0039fd9e308d15139f392e712589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.478ex; height:2.843ex;" alt="{\displaystyle x\lesssim y\iff \exists i\in I\colon x\in X_{i}\ni y\land x\lesssim _{i}y}"></span></dd></dl> <p>를 줄 수 있다. </p><p><b>컨트리먼 직선</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Countryman line</span>)은 다음 조건을 만족시키는 전순서 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05986ce35d21b64f59206cb040c9c1607dae82b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\leq )}"></span>이다. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>의 <a href="/wiki/%EC%A7%91%ED%95%A9%EC%9D%98_%ED%81%AC%EA%B8%B0" title="집합의 크기">집합의 크기</a>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}"></span>이다.</li> <li>임의의 양의 정수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/268db8293666fefd75cfb00513706171948edf09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.215ex; height:2.343ex;" alt="{\displaystyle X^{n}}"></span>은 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"></span>개의 전순서 집합들의 <a href="/wiki/%EB%B6%84%EB%A6%AC%ED%95%A9%EC%A7%91%ED%95%A9" title="분리합집합">분리합집합</a>과 순서 동형이다.</li></ul> <p>컨트리먼 직선의 존재는 <a href="/wiki/%EC%84%A0%ED%83%9D_%EA%B3%B5%EB%A6%AC" title="선택 공리">선택 공리</a>를 추가한 <a href="/wiki/%EC%B2%B4%EB%A5%B4%EB%A9%9C%EB%A1%9C-%ED%94%84%EB%A0%9D%EC%BC%88_%EC%A7%91%ED%95%A9%EB%A1%A0" title="체르멜로-프렝켈 집합론">체르멜로-프렝켈 집합론</a>만으로 보일 수 있으며, 이는 <a href="/wiki/%EC%82%AC%ED%95%98%EB%A1%A0_%EC%85%B8%EB%9D%BC%ED%9D%90" title="사하론 셸라흐">사하론 셸라흐</a>가 증명하였다.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="역사"><span id=".EC.97.AD.EC.82.AC"></span>역사</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=16" title="부분 편집: 역사"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>가산 조밀 전순서 집합의 분류 정리<sup id="cite_ref-Cantor_5-0" class="reference"><a href="#cite_note-Cantor-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:§9, 504–507</sup></span> 와 <a href="/wiki/%EB%B6%84%ED%95%B4_%EA%B0%80%EB%8A%A5_%EA%B3%B5%EA%B0%84" title="분해 가능 공간">분해 가능</a> <a href="/wiki/%EC%99%84%EB%B9%84_%EA%B2%A9%EC%9E%90" title="완비 격자">완비</a> 전순서 집합의 분류 정리<sup id="cite_ref-Cantor_5-1" class="reference"><a href="#cite_note-Cantor-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:§11, 510–512</sup></span>는 <a href="/wiki/%EA%B2%8C%EC%98%A4%EB%A5%B4%ED%81%AC_%EC%B9%B8%ED%86%A0%EC%96%B4" title="게오르크 칸토어">게오르크 칸토어</a>가 1895년에 증명하였다. </p> <div class="mw-heading mw-heading2"><h2 id="각주"><span id=".EA.B0.81.EC.A3.BC"></span>각주</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=17" title="부분 편집: 각주"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r35556958">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Geschke-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Geschke_1-0">가</a></sup> <sup><a href="#cite_ref-Geschke_1-1">나</a></sup></span> <span class="reference-text"><cite class="citation journal">Geschke, Stefan (2016). “Separable linear orders and universality” (영어). <a href="/wiki/ArXiv" title="ArXiv">arXiv</a>:<a rel="nofollow" class="external text" href="//arxiv.org/abs/1606.00338">1606.00338</a>. <a href="/wiki/%EB%B9%84%EB%B8%8C%EC%BD%94%EB%93%9C" title="비브코드">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/2016arXiv160600338G">2016arXiv160600338G</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Separable+linear+orders+and+universality&rft.date=2016&rft_id=info%3Aarxiv%2F1606.00338&rft_id=info%3Abibcode%2F2016arXiv160600338G&rft.aulast=Geschke&rft.aufirst=Stefan&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation book">Comtet, Louis (1974). 《Advanced combinatorics: The art of finite and infinite expansions》 (영어). Dordrecht: Reidel Publishing Company. <a href="/wiki/%EC%B2%B8%ED%8A%B8%EB%9E%84%EB%B8%94%EB%9D%BC%ED%8A%B8_%EB%A7%88%ED%8A%B8" title="첸트랄블라트 마트">Zbl</a> <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:0283.05001">0283.05001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+combinatorics%3A+The+art+of+finite+and+infinite+expansions&rft.place=Dordrecht&rft.pub=Reidel+Publishing+Company&rft.date=1974&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0283.05001&rft.aulast=Comtet&rft.aufirst=Louis&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><cite class="citation book">Just, Winfried; Weese, Martin (1997). <a rel="nofollow" class="external text" href="http://bookstore.ams.org/gsm-18/">《Discovering modern set theory II: set-theoretic tools for every mathematician》</a>. Graduate Studies in Mathematics (영어) <b>18</b>. American Mathematical Society. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-0-8218-0528-2" title="특수:책찾기/978-0-8218-0528-2"><bdi>978-0-8218-0528-2</bdi></a>. <a href="/wiki/%EC%B2%B8%ED%8A%B8%EB%9E%84%EB%B8%94%EB%9D%BC%ED%8A%B8_%EB%A7%88%ED%8A%B8" title="첸트랄블라트 마트">Zbl</a> <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:0887.03036">0887.03036</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discovering+modern+set+theory+II%3A+set-theoretic+tools+for+every+mathematician&rft.series=Graduate+Studies+in+Mathematics&rft.pub=American+Mathematical+Society&rft.date=1997&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0887.03036&rft.isbn=978-0-8218-0528-2&rft.aulast=Just&rft.aufirst=Winfried&rft.au=Weese%2C+Martin&rft_id=http%3A%2F%2Fbookstore.ams.org%2Fgsm-18%2F&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><cite class="citation journal"><a href="/wiki/%EC%82%AC%ED%95%98%EB%A1%A0_%EC%85%B8%EB%9D%BC%ED%9D%90" title="사하론 셸라흐">Shelah, Saharon</a> (1976년 7월). “Decomposing uncountable squares to countably many chains”. 《Journal of Combinatorial Theory Series A》 (영어) <b>21</b> (1): 110–114. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016%2F0097-3165%2876%2990053-4">10.1016/0097-3165(76)90053-4</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0097-3165">0097-3165</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Combinatorial+Theory+Series+A&rft.atitle=Decomposing+uncountable+squares+to+countably+many+chains&rft.volume=21&rft.issue=1&rft.pages=110-114&rft.date=1976-07&rft_id=info%3Adoi%2F10.1016%2F0097-3165%2876%2990053-4&rft.issn=0097-3165&rft.aulast=Shelah&rft.aufirst=Saharon&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Cantor-5"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Cantor_5-0">가</a></sup> <sup><a href="#cite_ref-Cantor_5-1">나</a></sup></span> <span class="reference-text"><cite class="citation journal"><a href="/wiki/%EA%B2%8C%EC%98%A4%EB%A5%B4%ED%81%AC_%EC%B9%B8%ED%86%A0%EC%96%B4" title="게오르크 칸토어">Cantor, Georg</a> (1895). <a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN00225557X">“Beiträge zur Begründung der transfiniten Mengenlehre (Erster Artikel)”</a>. 《Mathematische Annalen》 (독일어) <b>46</b> (4): 481–512. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2Fbf02124929">10.1007/bf02124929</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0025-5831">0025-5831</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Beitr%C3%A4ge+zur+Begr%C3%BCndung+der+transfiniten+Mengenlehre+%28Erster+Artikel%29&rft.volume=46&rft.issue=4&rft.pages=481-512&rft.date=1895&rft_id=info%3Adoi%2F10.1007%2Fbf02124929&rft.issn=0025-5831&rft.aulast=Cantor&rft.aufirst=Georg&rft_id=http%3A%2F%2Fwww.digizeitschriften.de%2Fdms%2Fresolveppn%2F%3FPID%3DGDZPPN00225557X&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="외부_링크"><span id=".EC.99.B8.EB.B6.80_.EB.A7.81.ED.81.AC"></span>외부 링크</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9&action=edit&section=18" title="부분 편집: 외부 링크"><span>편집</span></a><span 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href="https://commons.wikimedia.org/wiki/Category:Weak_ordering?uselang=ko">원전순서</a></b></div></div></div> </div> <ul><li><cite class="citation web"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Totally_ordered_set">“Totally ordered set”</a>. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-1-55608-010-4" title="특수:책찾기/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Totally+ordered+set&rft.date=2001&rft.isbn=978-1-55608-010-4&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FTotally_ordered_set&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TotalOrder.html">“Total order”</a>. 《<a href="/wiki/%EB%A7%A4%EC%8A%A4%EC%9B%94%EB%93%9C" title="매스월드">Wolfram MathWorld</a>》 (영어). Wolfram Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Total+order&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTotalOrder.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TotallyOrderedSet.html">“Totally ordered set”</a>. 《<a href="/wiki/%EB%A7%A4%EC%8A%A4%EC%9B%94%EB%93%9C" title="매스월드">Wolfram MathWorld</a>》 (영어). Wolfram Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Totally+ordered+set&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTotallyOrderedSet.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/total+order">“Total order”</a>. 《nLab》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=nLab&rft.atitle=Total+order&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Ftotal%2Border&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/linear+order">“Linear order”</a>. 《nLab》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=nLab&rft.atitle=Linear+order&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Flinear%2Border&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Definition:Total_Ordering">“Definition: total ordering”</a>. 《ProofWiki》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ProofWiki&rft.atitle=Definition%3A+total+ordering&rft_id=https%3A%2F%2Fproofwiki.org%2Fwiki%2FDefinition%3ATotal_Ordering&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Definition:Totally_Ordered_Set">“Definition: totally ordered set”</a>. 《ProofWiki》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ProofWiki&rft.atitle=Definition%3A+totally+ordered+set&rft_id=https%3A%2F%2Fproofwiki.org%2Fwiki%2FDefinition%3ATotally_Ordered_Set&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="http://mathoverflow.net/questions/37272/are-all-sets-totally-ordered">“Are all sets totally ordered?”</a> (영어). Math Overflow.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Are+all+sets+totally+ordered%3F&rft.pub=Math+Overflow&rft_id=http%3A%2F%2Fmathoverflow.net%2Fquestions%2F37272%2Fare-all-sets-totally-ordered&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EC%A0%84%EC%88%9C%EC%84%9C+%EC%A7%91%ED%95%A9" class="Z3988"><span style="display:none;"> </span></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">원본 주소 "<a dir="ltr" href="https://ko.wikipedia.org/w/index.php?title=전순서_집합&oldid=38071227">https://ko.wikipedia.org/w/index.php?title=전순서_집합&oldid=38071227</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%B6%84%EB%A5%98" 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