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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=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" 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=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" 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=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" 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=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" 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-메타콤팩트_공간" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#메타콤팩트_공간"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>메타콤팩트 공간</span> </div> </a> <ul id="toc-메타콤팩트_공간-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-가산_파라콤팩트_공간" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#가산_파라콤팩트_공간"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>가산 파라콤팩트 공간</span> </div> </a> <ul id="toc-가산_파라콤팩트_공간-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-콤팩트성과의_관계" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#콤팩트성과의_관계"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>콤팩트성과의 관계</span> </div> </a> <ul id="toc-콤팩트성과의_관계-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-파라콤팩트_하우스도르프_공간" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#파라콤팩트_하우스도르프_공간"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>파라콤팩트 하우스도르프 공간</span> </div> </a> <ul id="toc-파라콤팩트_하우스도르프_공간-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">3</span> <span>예</span> </div> </a> <ul id="toc-예-sublist" class="vector-toc-list"> </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">4</span> <span>역사</span> </div> </a> <ul id="toc-역사-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-참고_문헌" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#참고_문헌"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>참고 문헌</span> </div> </a> <ul id="toc-참고_문헌-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">6</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">파라콤팩트 공간</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. 15개 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-15" 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">15개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_paracompacte" title="Espai paracompacte – 카탈로니아어" lang="ca" hreflang="ca" data-title="Espai paracompacte" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Parakompakter_Raum" title="Parakompakter Raum – 독일어" lang="de" hreflang="de" data-title="Parakompakter Raum" 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/Paracompact_space" title="Paracompact space – 영어" lang="en" hreflang="en" data-title="Paracompact space" 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/Parakompakta_spaco" title="Parakompakta spaco – 에스페란토어" lang="eo" hreflang="eo" data-title="Parakompakta spaco" 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/Espacio_paracompacto" title="Espacio paracompacto – 스페인어" lang="es" hreflang="es" data-title="Espacio paracompacto" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D9%BE%DB%8C%D8%B1%D8%A7%D9%81%D8%B4%D8%B1%D8%AF%D9%87" title="فضای پیرافشرده – 페르시아어" lang="fa" hreflang="fa" data-title="فضای پیرافشرده" 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/Espace_paracompact" title="Espace paracompact – 프랑스어" lang="fr" hreflang="fr" data-title="Espace paracompact" data-language-autonym="Français" data-language-local-name="프랑스어" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_paracompatto" title="Spazio paracompatto – 이탈리아어" lang="it" hreflang="it" data-title="Spazio paracompatto" data-language-autonym="Italiano" data-language-local-name="이탈리아어" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%91%E3%83%A9%E3%82%B3%E3%83%B3%E3%83%91%E3%82%AF%E3%83%88%E7%A9%BA%E9%96%93" title="パラコンパクト空間 – 일본어" lang="ja" hreflang="ja" data-title="パラコンパクト空間" data-language-autonym="日本語" data-language-local-name="일본어" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Paracompacte_ruimte" title="Paracompacte ruimte – 네덜란드어" lang="nl" hreflang="nl" data-title="Paracompacte ruimte" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_parazwarta" title="Przestrzeń parazwarta – 폴란드어" lang="pl" hreflang="pl" data-title="Przestrzeń parazwarta" data-language-autonym="Polski" data-language-local-name="폴란드어" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_paracompacto" title="Espaço paracompacto – 포르투갈어" lang="pt" hreflang="pt" data-title="Espaço paracompacto" data-language-autonym="Português" data-language-local-name="포르투갈어" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%BA%D0%BE%D0%BC%D0%BF%D0%B0%D0%BA%D1%82%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Паракомпактное пространство – 러시아어" lang="ru" hreflang="ru" data-title="Паракомпактное пространство" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D0%B0%D0%BA%D0%BE%D0%BC%D0%BF%D0%B0%D0%BA%D1%82%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Паракомпактний простір – 우크라이나어" lang="uk" hreflang="uk" data-title="Паракомпактний простір" data-language-autonym="Українська" data-language-local-name="우크라이나어" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BB%BF%E7%B4%A7%E7%A9%BA%E9%97%B4" title="仿紧空间 – 중국어" lang="zh" hreflang="zh" data-title="仿紧空间" data-language-autonym="中文" data-language-local-name="중국어" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q970119#sitelinks-wikipedia" title="언어 간 링크 편집" class="wbc-editpage">링크 편집</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="이름공간"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><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="본문 보기 [c]" accesskey="c"><span>문서</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/%ED%86%A0%EB%A1%A0:%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" rel="discussion" title="문서의 내용에 대한 토론 문서 [t]" accesskey="t"><span>토론</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="언어 변종 바꾸기" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" 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class="vector-appearance-landmark" aria-label="보이기"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">보이기</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">숨기기</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">위키백과, 우리 모두의 백과사전.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(<a href="/w/index.php?title=%EB%A9%94%ED%83%80%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84&redirect=no" class="mw-redirect" title="메타콤팩트 공간">메타콤팩트 공간</a>에서 넘어옴)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ko" dir="ltr"><p><span class="nowrap"></span> <a href="/wiki/%EC%9D%BC%EB%B0%98%EC%9C%84%EC%83%81%EC%88%98%ED%95%99" title="일반위상수학">일반위상수학</a>에서 <b>파라콤팩트 공간</b>(paracompact空間, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">paracompact space</span>)은 <a href="/wiki/%EB%8B%A8%EC%9C%84_%EB%B6%84%ED%95%A0" title="단위 분할">단위 분할</a>의 존재를 증명하기 위하여 필요한, <a href="/wiki/%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="콤팩트 공간">콤팩트 공간</a>의 개념의 일반화이다. 수학에서 흔히 사용되는 대부분의 공간은 파라콤팩트 공간이며, 파라콤팩트성을 가정하면 <a href="/wiki/%EB%8B%A8%EC%9C%84_%EB%B6%84%ED%95%A0" title="단위 분할">단위 분할</a>을 통해 <a href="/wiki/%ED%95%B4%EC%84%9D%ED%95%99_(%EC%88%98%ED%95%99)" title="해석학 (수학)">해석학</a>적 구조를 쉽게 정의할 수 있다. </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=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84&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%9C%84%EC%83%81_%EA%B3%B5%EA%B0%84_(%EC%88%98%ED%95%99)" title="위상 공간 (수학)">위상 공간</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}"> <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> 위의 <a href="/wiki/%EC%A7%91%ED%95%A9%EC%A1%B1" title="집합족">집합족</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 \{U_{i}\}_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <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 \{U_{i}\}_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/710ee8280766ae621336e9e60afb939d27e841a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.437ex; height:2.843ex;" alt="{\displaystyle \{U_{i}\}_{i\in I}}"></span>가 주어졌다고 하자. 만약 다음 조건을 만족시키는 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</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 \{V_{j}\}_{j\in J}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mi>J</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{V_{j}\}_{j\in J}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f63719fbc2989013d6a53488e69c9616cd7bf2eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.636ex; height:3.009ex;" alt="{\displaystyle \{V_{j}\}_{j\in J}}"></span>가 존재한다면, <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 \{U_{i}\}_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <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 \{U_{i}\}_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/710ee8280766ae621336e9e60afb939d27e841a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.437ex; height:2.843ex;" alt="{\displaystyle \{U_{i}\}_{i\in I}}"></span>를 <b>국소 유한 집합족</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">locally finite family of sets</span>)이라고 한다.<sup id="cite_ref-조용승_1-0" class="reference"><a href="#cite_note-조용승-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:68</sup></span> </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 j\in J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>∈</mo> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\in J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6d6a463cc24fcc2d0ad452450c7a9d845409a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.297ex; height:2.509ex;" alt="{\displaystyle j\in J}"></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 \{i\in I\colon U_{i}\cap V_{j}\neq \varnothing \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo>:</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∩</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{i\in I\colon U_{i}\cap V_{j}\neq \varnothing \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a9e08e16eb873e48b992246458116f77d900448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.315ex; height:3.009ex;" alt="{\displaystyle \{i\in I\colon U_{i}\cap V_{j}\neq \varnothing \}}"></span>는 <a href="/wiki/%EC%9C%A0%ED%95%9C_%EC%A7%91%ED%95%A9" title="유한 집합">유한 집합</a>이다.</li></ul> <p>즉, 국소 유한 집합족은 모든 점에서 유한 개의 집합족 원소들과 겹치는 근방을 잡을 수 있는 집합족이다. </p><p><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> <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> 위의 임의의 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a>에 대하여 국소 유한 열린 덮개인 <a href="/wiki/%EC%84%B8%EB%B6%84_(%EC%9C%84%EC%83%81%EC%88%98%ED%95%99)" class="mw-redirect" title="세분 (위상수학)">세분</a>을 찾을 수 있다면, <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>를 <b>파라콤팩트 공간</b>이라고 한다.<sup id="cite_ref-조용승_1-1" class="reference"><a href="#cite_note-조용승-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:68</sup></span> </p> <div class="mw-heading mw-heading3"><h3 id="메타콤팩트_공간"><span id=".EB.A9.94.ED.83.80.EC.BD.A4.ED.8C.A9.ED.8A.B8_.EA.B3.B5.EA.B0.84"></span>메타콤팩트 공간</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84&action=edit&section=2" title="부분 편집: 메타콤팩트 공간"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>파라콤팩트 공간의 정의를 변형시켜 다음과 같은 개념들을 정의할 수 있다. </p> <ul><li><b>메조콤팩트 공간</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">mesocompact space</span>)</li> <li><b>메타콤팩트 공간</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">metacompact space</span>)</li> <li><b>직교 콤팩트 공간</b>(直交-, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">orthocompact space</span>)</li></ul> <p><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> <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%A1%B1" title="집합족">집합족</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 \{U_{i}\}_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <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 \{U_{i}\}_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/710ee8280766ae621336e9e60afb939d27e841a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.437ex; height:2.843ex;" alt="{\displaystyle \{U_{i}\}_{i\in I}}"></span>가 다음 조건을 만족시키면, <b>콤팩트 유한 집합족</b>(compact有限-, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">compact-finite family of sets</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>:23</sup></span> </p> <ul><li>임의의 <a href="/wiki/%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EC%A7%91%ED%95%A9" class="mw-redirect" title="콤팩트 집합">콤팩트 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 K\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba18595612961c1a8de8fbb2464e9bcfd4554da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.144ex; height:2.343ex;" alt="{\displaystyle K\subseteq 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 \{i\in I\colon K\cap U_{i}\neq \varnothing \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo>:</mo> <mi>K</mi> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≠</mo> <mi class="MJX-variant">∅</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{i\in I\colon K\cap U_{i}\neq \varnothing \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7072afad005099d914c8a543fa47c0a31ad78f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.116ex; height:2.843ex;" alt="{\displaystyle \{i\in I\colon K\cap U_{i}\neq \varnothing \}}"></span>는 <a href="/wiki/%EC%9C%A0%ED%95%9C_%EC%A7%91%ED%95%A9" title="유한 집합">유한 집합</a>이다.</li></ul> <p>즉, 콤팩트 유한 집합족은 모든 콤팩트 집합이 유한 개의 집합족 원소와 만나는 <a href="/wiki/%EC%A7%91%ED%95%A9%EC%A1%B1" title="집합족">집합족</a>이다. </p><p><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> <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%A1%B1" title="집합족">집합족</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 \{U_{i}\}_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <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 \{U_{i}\}_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/710ee8280766ae621336e9e60afb939d27e841a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.437ex; height:2.843ex;" alt="{\displaystyle \{U_{i}\}_{i\in I}}"></span>가 다음 조건을 만족시키면, <b>점 유한 집합족</b>(點有限-, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">point-finite family of sets</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\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 \{i\in I\colon x\in U_{i}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{i\in I\colon x\in U_{i}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4338bcb1bbd0f51d59822b1d602aba614b08c3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.731ex; height:2.843ex;" alt="{\displaystyle \{i\in I\colon x\in U_{i}\}}"></span>는 <a href="/wiki/%EC%9C%A0%ED%95%9C_%EC%A7%91%ED%95%A9" title="유한 집합">유한 집합</a>이다.</li></ul> <p>즉, 점 유한 집합족은 모든 점이 유한 개의 집합족 원소에만 포함되는 <a href="/wiki/%EC%A7%91%ED%95%A9%EC%A1%B1" title="집합족">집합족</a>이다. </p><p><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> <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%97%B4%EB%A6%B0%EC%A7%91%ED%95%A9" title="열린집합">열린집합</a>들의 <a href="/wiki/%EC%A7%91%ED%95%A9%EC%A1%B1" title="집합족">집합족</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 \{U_{i}\}_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <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 \{U_{i}\}_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/710ee8280766ae621336e9e60afb939d27e841a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.437ex; height:2.843ex;" alt="{\displaystyle \{U_{i}\}_{i\in I}}"></span>가 다음 조건을 만족시키면, <b>내부 보존 집합족</b>(內部保存-, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">interior-preserving family of sets</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\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 \bigcap \{U_{i}\colon i\in I,\,x\in U_{i}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋂</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>x</mi> <mo>∈</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcap \{U_{i}\colon i\in I,\,x\in U_{i}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1ee45c4775c4851006be4a5463fc2bd90c6ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.121ex; height:3.843ex;" alt="{\displaystyle \bigcap \{U_{i}\colon i\in I,\,x\in U_{i}\}}"></span>는 <a href="/wiki/%EC%97%B4%EB%A6%B0%EC%A7%91%ED%95%A9" title="열린집합">열린집합</a>이다.</li></ul> <p>이 개념들로부터, 다음과 같은 꼴의 정의를 내릴 수 있다. </p> <dl><dd><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> <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> 위의 임의의 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a>가 조건 P를 만족시키는 열린 <a href="/wiki/%EC%84%B8%EB%B6%84_(%EC%9C%84%EC%83%81%EC%88%98%ED%95%99)" class="mw-redirect" title="세분 (위상수학)">세분</a>을 갖는다면, <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>를 <b>~ 공간</b>이라고 한다.</dd></dl> <p>이 정의들은 다음과 같다. </p> <table class="wikitable"> <tbody><tr> <th>개념</th> <th><a href="/wiki/%EC%84%B8%EB%B6%84_(%EC%9C%84%EC%83%81%EC%88%98%ED%95%99)" class="mw-redirect" title="세분 (위상수학)">세분</a>의 조건 </th></tr> <tr> <th>파라콤팩트 공간 </th> <td>국소 유한 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a> </td></tr> <tr> <th>메조콤팩트 공간 </th> <td>콤팩트 유한 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a><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>:200</sup></span> </td></tr> <tr> <th>메타콤팩트 공간 </th> <td>점 유한 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a> </td></tr> <tr> <th>직교 콤팩트 공간 </th> <td>내부 보존 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="가산_파라콤팩트_공간"><span id=".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"></span>가산 파라콤팩트 공간</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84&action=edit&section=3" title="부분 편집: 가산 파라콤팩트 공간"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>파라콤팩트·메조콤팩트·메타콤팩트·직교 콤팩트 공간의 정의에서, “임의의 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a>”를 “<a href="/wiki/%EA%B0%80%EC%82%B0_%EC%A7%91%ED%95%A9" title="가산 집합">가산</a> <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a>”로 약화시키면 </p> <ul><li><b>가산 파라콤팩트 공간</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">countably paracompact space</span>)</li> <li><b>가산 메조콤팩트 공간</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">countably mesocompact space</span>)</li> <li><b>가산 메타콤팩트 공간</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">countably metacompact space</span>)</li> <li><b>가산 직교 콤팩트 공간</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">countably orthoparacompact space</span>)</li></ul> <p>의 개념을 얻는다. 예를 들어, <b>가산 파라콤팩트 공간</b>은 모든 <a href="/wiki/%EA%B0%80%EC%82%B0_%EC%A7%91%ED%95%A9" title="가산 집합">가산</a> <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a>가 국소 유한 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a>인 <a href="/wiki/%EC%84%B8%EB%B6%84_(%EC%9C%84%EC%83%81%EC%88%98%ED%95%99)" class="mw-redirect" title="세분 (위상수학)">세분</a>을 갖는 <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>이다. </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=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84&action=edit&section=4" title="부분 편집: 성질"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>~콤팩트 공간과 <a href="/wiki/%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="콤팩트 공간">콤팩트 공간</a>의 <a href="/wiki/%EA%B3%B1%EA%B3%B5%EA%B0%84" class="mw-redirect" title="곱공간">곱공간</a>에 대하여 다음이 성립한다. </p> <ul><li>콤팩트 공간과 파라콤팩트 공간의 <a href="/wiki/%EA%B3%B1%EA%B3%B5%EA%B0%84" class="mw-redirect" title="곱공간">곱공간</a>은 파라콤팩트 공간이다.<sup id="cite_ref-Munkres_4-0" class="reference"><a href="#cite_note-Munkres-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:260</sup></span></li> <li>콤팩트 공간과 메조콤팩트 공간의 <a href="/wiki/%EA%B3%B1%EA%B3%B5%EA%B0%84" class="mw-redirect" title="곱공간">곱공간</a>은 메조콤팩트 공간이다.</li> <li>콤팩트 공간과 메타콤팩트 공간의 <a href="/wiki/%EA%B3%B1%EA%B3%B5%EA%B0%84" class="mw-redirect" title="곱공간">곱공간</a>은 메타콤팩트 공간이다.</li> <li>콤팩트 공간과 가산 파라콤팩트 공간의 <a href="/wiki/%EA%B3%B1%EA%B3%B5%EA%B0%84" class="mw-redirect" title="곱공간">곱공간</a>은 가산 파라콤팩트 공간이다.<sup id="cite_ref-Willard_5-0" class="reference"><a href="#cite_note-Willard-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:159, 21A.3</sup></span></li></ul> <p>그러나 직교 콤팩트 공간의 경우 이러한 꼴의 정리가 성립하지 않는다. 이에 대한 부분적인 결과인 <b>스콧 정리</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Scott’s theorem</span>)에 따르면, 임의의 직교 콤팩트 공간 <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 대하여 다음 두 조건이 서로 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>이다.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </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\times [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <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 X\times [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd76f8aa27cdb748a869ec11fc6548bcab6d396b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.473ex; height:2.843ex;" alt="{\displaystyle X\times [0,1]}"></span>은 직교 콤팩트 공간이다.</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 가산 메타콤팩트 공간이다.</li></ul> <p>또한, ~콤팩트 공간의 <a href="/wiki/%EB%8B%AB%ED%9E%8C%EC%A7%91%ED%95%A9" class="mw-redirect" title="닫힌집합">닫힌집합</a>에 대하여 다음이 성립한다. </p> <ul><li>파라콤팩트 공간의 <a href="/wiki/%EB%8B%AB%ED%9E%8C%EC%A7%91%ED%95%A9" class="mw-redirect" title="닫힌집합">닫힌집합</a>은 파라콤팩트 공간이다.<sup id="cite_ref-Munkres_4-1" class="reference"><a href="#cite_note-Munkres-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:254</sup></span></li> <li>메조콤팩트 공간의 <a href="/wiki/%EB%8B%AB%ED%9E%8C%EC%A7%91%ED%95%A9" class="mw-redirect" title="닫힌집합">닫힌집합</a>은 메조콤팩트 공간이다.</li> <li>메타콤팩트 공간의 <a href="/wiki/%EB%8B%AB%ED%9E%8C%EC%A7%91%ED%95%A9" class="mw-redirect" title="닫힌집합">닫힌집합</a>은 메타콤팩트 공간이다.</li> <li>직교 콤팩트 공간의 <a href="/wiki/%EB%8B%AB%ED%9E%8C%EC%A7%91%ED%95%A9" class="mw-redirect" title="닫힌집합">닫힌집합</a>은 직교 콤팩트 공간이다.</li> <li>가산 파라콤팩트 공간의 <a href="/wiki/%EB%8B%AB%ED%9E%8C%EC%A7%91%ED%95%A9" class="mw-redirect" title="닫힌집합">닫힌집합</a>은 가산 파라콤팩트 공간이다.<sup id="cite_ref-Willard_5-1" class="reference"><a href="#cite_note-Willard-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:159, 21A.2</sup></span></li></ul> <p>한편, 일반적으로 파라콤팩트 공간의 임의의 부분공간은 파라콤팩트 공간이 되지 않으므로 파라콤팩트성은 <a href="/wiki/%EC%9C%A0%EC%A0%84%EC%A0%81_%EC%84%B1%EC%A7%88" class="mw-redirect" title="유전적 성질">유전적 성질</a>이 아니다. 또한, <a href="/wiki/%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="콤팩트 공간">콤팩트 공간</a>들을 모으면 <a href="/wiki/%ED%8B%B0%ED%98%B8%EB%85%B8%ED%94%84_%EC%A0%95%EB%A6%AC" title="티호노프 정리">티호노프 정리</a>에 의해 그 곱공간 역시 <a href="/wiki/%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="콤팩트 공간">콤팩트 공간</a>이 되는 것과는 다르게, 파라콤팩트 공간의 임의의 <a href="/wiki/%EA%B3%B1%EA%B3%B5%EA%B0%84" class="mw-redirect" title="곱공간">곱공간</a>은 파라콤팩트 공간이 되지 않는다.<sup id="cite_ref-Munkres_4-2" class="reference"><a href="#cite_note-Munkres-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:253</sup></span> </p> <div class="mw-heading mw-heading3"><h3 id="콤팩트성과의_관계"><span id=".EC.BD.A4.ED.8C.A9.ED.8A.B8.EC.84.B1.EA.B3.BC.EC.9D.98_.EA.B4.80.EA.B3.84"></span>콤팩트성과의 관계</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84&action=edit&section=5" title="부분 편집: 콤팩트성과의 관계"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>다음과 같은 포함 관계가 성립한다. </p> <dl><dd><table style="text-align: center"> <tbody><tr> <td><a href="/wiki/%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="콤팩트 공간">콤팩트 공간</a></td> <td>→</td> <td>파라콤팩트 공간</td> <td>→</td> <td>메조콤팩트 공간</td> <td>→</td> <td>메타콤팩트 공간</td> <td>→</td> <td>직교 콤팩트 공간 </td></tr> <tr> <td>↓</td> <td></td> <td>↓</td> <td></td> <td>↓</td> <td></td> <td>↓</td> <td></td> <td>↓ </td></tr> <tr> <td><a href="/wiki/%EA%B0%80%EC%82%B0_%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="가산 콤팩트 공간">가산 콤팩트 공간</a></td> <td>→</td> <td>가산 파라콤팩트 공간</td> <td>→</td> <td>가산 메조콤팩트 공간</td> <td>→</td> <td>가산 메타콤팩트 공간</td> <td>→</td> <td>가산 직교 콤팩트 공간 </td></tr></tbody></table></dd></dl> <p>이 밖에도, 다음과 같은 함의 관계가 성립한다. </p> <ul><li>파라콤팩트 <a href="/wiki/%ED%9D%AC%EB%B0%95_%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="희박 콤팩트 공간">희박 콤팩트 공간</a>은 콤팩트 공간이다.</li> <li>메타콤팩트 <a href="/wiki/%EA%B0%80%EC%82%B0_%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="가산 콤팩트 공간">가산 콤팩트 공간</a>은 콤팩트 공간이다.</li> <li>파라콤팩트 <a href="/wiki/%EC%A0%95%EC%B9%99_%EA%B3%B5%EA%B0%84" title="정칙 공간">정칙 공간</a>은 <a href="/wiki/%EC%A4%80%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="준파라콤팩트 공간">준파라콤팩트 공간</a>이다.</li> <li>모든 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a>가 국소 유한 <a href="/wiki/%EC%84%B8%EB%B6%84_(%EC%9C%84%EC%83%81%EC%88%98%ED%95%99)" class="mw-redirect" title="세분 (위상수학)">세분</a>을 갖는 <a href="/wiki/%EC%A0%95%EC%B9%99_%EA%B3%B5%EA%B0%84" title="정칙 공간">정칙 공간</a>은 파라콤팩트 공간이다.</li> <li>(<b><a href="/wiki/%EB%AA%A8%EB%A6%AC%ED%83%80_%EC%A0%95%EB%A6%AC" class="mw-redirect" title="모리타 정리">모리타 정리</a></b> <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Morita’s theorem</span>) <a href="/wiki/%EC%A0%95%EC%B9%99_%EA%B3%B5%EA%B0%84" title="정칙 공간">정칙</a> <a href="/wiki/%EB%A6%B0%EB%8D%B8%EB%A2%B0%ED%94%84_%EA%B3%B5%EA%B0%84" title="린델뢰프 공간">린델뢰프 공간</a>은 파라콤팩트 공간이다.<sup id="cite_ref-Munkres_4-3" class="reference"><a href="#cite_note-Munkres-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:257</sup></span><sup id="cite_ref-Morita_7-0" class="reference"><a href="#cite_note-Morita-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> 특히, 모든 <a href="/wiki/%EA%B5%AD%EC%86%8C_%EC%BD%A4%ED%8C%A9%ED%8A%B8" class="mw-redirect" title="국소 콤팩트">국소 콤팩트</a> <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%A0%9C2_%EA%B0%80%EC%82%B0_%EA%B3%B5%EA%B0%84" title="제2 가산 공간">제2 가산 공간</a>은 파라콤팩트 공간이다. 반대로, <a href="/wiki/%EA%B0%80%EC%82%B0_%EA%B0%95%ED%95%98%ED%96%A5_%EB%B0%98%EC%82%AC%EC%8A%AC_%EC%A1%B0%EA%B1%B4" class="mw-redirect" title="가산 강하향 반사슬 조건">가산 강하향 반사슬 조건</a>을 만족시키는 파라콤팩트 공간은 <a href="/wiki/%EB%A6%B0%EB%8D%B8%EB%A2%B0%ED%94%84_%EA%B3%B5%EA%B0%84" title="린델뢰프 공간">린델뢰프 공간</a>이다. 특히, <a href="/wiki/%EB%B6%84%ED%95%B4_%EA%B0%80%EB%8A%A5" class="mw-redirect" title="분해 가능">분해 가능</a> 파라콤팩트 공간은 <a href="/wiki/%EB%A6%B0%EB%8D%B8%EB%A2%B0%ED%94%84_%EA%B3%B5%EA%B0%84" title="린델뢰프 공간">린델뢰프 공간</a>이다.</li> <li><a href="/wiki/%EA%B5%AD%EC%86%8C_%EC%BD%A4%ED%8C%A9%ED%8A%B8" class="mw-redirect" title="국소 콤팩트">국소 콤팩트</a> <a href="/wiki/%EC%97%B0%EA%B2%B0_%EA%B3%B5%EA%B0%84" title="연결 공간">연결</a> <a href="/wiki/%EC%9C%84%EC%83%81%EA%B5%B0" title="위상군">위상군</a>은 파라콤팩트 공간이다.<sup id="cite_ref-Munkres_4-4" class="reference"><a href="#cite_note-Munkres-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:261</sup></span></li> <li><a href="/wiki/%EC%99%84%EC%A0%84_%EC%A0%95%EA%B7%9C_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="완전 정규 공간">완전 정규 공간</a>은 가산 파라콤팩트 공간이다.<sup id="cite_ref-Willard_5-2" class="reference"><a href="#cite_note-Willard-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:159, 21A.1</sup></span></li> <li><a href="/wiki/%EC%88%9C%EC%84%9C_%EC%9C%84%EC%83%81" title="순서 위상">순서 위상</a>을 갖춘 <a href="/wiki/%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9" title="전순서 집합">전순서 집합</a>의 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>은 직교 콤팩트 공간이자 가산 파라콤팩트 공간이다.<sup id="cite_ref-Künzi_8-0" class="reference"><a href="#cite_note-Künzi-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:17</sup></span></li> <li><a href="/wiki/%EC%88%9C%EC%84%9C_%EC%9C%84%EC%83%81" title="순서 위상">순서 위상</a>을 갖춘 <a href="/wiki/%EC%A0%84%EC%88%9C%EC%84%9C_%EC%A7%91%ED%95%A9" title="전순서 집합">전순서 집합</a>이 메타콤팩트 공간이라면, 파라콤팩트 공간이다.<sup id="cite_ref-Gulden_9-0" class="reference"><a href="#cite_note-Gulden-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:199, Theorem 1</sup></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="파라콤팩트_하우스도르프_공간"><span id=".ED.8C.8C.EB.9D.BC.EC.BD.A4.ED.8C.A9.ED.8A.B8_.ED.95.98.EC.9A.B0.EC.8A.A4.EB.8F.84.EB.A5.B4.ED.94.84_.EA.B3.B5.EA.B0.84"></span>파라콤팩트 하우스도르프 공간</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84&action=edit&section=6" title="부분 편집: 파라콤팩트 하우스도르프 공간"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>파라콤팩트 공간에 <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 공간</a>의 조건을 추가하면, 여러 유용한 성질들이 성립한다. (이 때문에, 일부 문헌에서는 모든 파라콤팩트 공간이 하우스도르프 공간이 되게 정의한다.) 이 가운데 가장 중요한 것인 <b>디외도네 정리</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Dieudonne’s theorem</span>)에 따르면, 모든 파라콤팩트 <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 공간</a>은 <a href="/wiki/%EC%A0%95%EA%B7%9C_%EA%B3%B5%EA%B0%84" title="정규 공간">정규 공간</a>이다.<sup id="cite_ref-Munkres_4-5" class="reference"><a href="#cite_note-Munkres-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:253</sup></span> <a href="/wiki/%EB%AA%A8%EB%A6%AC%ED%83%80_%EC%A0%95%EB%A6%AC" class="mw-redirect" title="모리타 정리">모리타 정리</a>와 디외도네 정리로부터, <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/%EB%A6%B0%EB%8D%B8%EB%A2%B0%ED%94%84_%EA%B3%B5%EA%B0%84" title="린델뢰프 공간">린델뢰프 공간</a>에 대하여 다음 조건들이 서로 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>임을 알 수 있다. </p> <ul><li><a href="/wiki/%EC%A0%95%EC%B9%99_%EA%B3%B5%EA%B0%84" title="정칙 공간">정칙 공간</a>이다.</li> <li><a href="/wiki/%EC%A0%95%EA%B7%9C_%EA%B3%B5%EA%B0%84" title="정규 공간">정규 공간</a>이다.</li> <li>파라콤팩트 공간이다.</li></ul> <p><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 공간</a>에 대하여, 다음 조건들이 서로 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>이다. </p> <ul><li>파라콤팩트 공간이다.</li> <li>임의의 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a>에 대하여, 이에 종속되는 <a href="/wiki/%EB%8B%A8%EC%9C%84_%EB%B6%84%ED%95%A0" title="단위 분할">단위 분할</a>이 존재한다.</li> <li>모든 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a>는 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린</a> <a href="/wiki/%EC%84%B1%ED%98%95_%EC%84%B8%EB%B6%84" class="mw-redirect" title="성형 세분">성형 세분</a>을 갖는다.<sup id="cite_ref-Willard_5-3" class="reference"><a href="#cite_note-Willard-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:151, Corollary 20.15</sup></span></li> <li>모든 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린 덮개</a>는 <a href="/wiki/%EC%97%B4%EB%A6%B0_%EB%8D%AE%EA%B0%9C" class="mw-redirect" title="열린 덮개">열린</a> <a href="/wiki/%EC%84%B1%ED%98%95_%EC%84%B8%EB%B6%84" class="mw-redirect" title="성형 세분">무게 중심 세분</a>을 갖는다.<sup id="cite_ref-Willard_5-4" class="reference"><a href="#cite_note-Willard-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:149, Theorem 20.14</sup></span></li></ul> <p>따라서, 파라콤팩트성은 <a href="/wiki/%EB%AF%B8%EB%B6%84%EA%B8%B0%ED%95%98%ED%95%99" title="미분기하학">미분기하학</a>에서 핵심적인 단위 분할의 개념과 밀접하게 연관되어 있다. </p><p>또한, <a href="/wiki/%EC%8A%A4%EB%AF%B8%EB%A5%B4%EB%85%B8%ED%94%84_%EA%B1%B0%EB%A6%AC%ED%99%94_%EC%A0%95%EB%A6%AC" class="mw-redirect" title="스미르노프 거리화 정리">스미르노프 거리화 정리</a>에 따르면, 임의의 위상 공간에 대하여 다음 두 조건이 서로 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>이다.<sup id="cite_ref-Munkres_4-6" class="reference"><a href="#cite_note-Munkres-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:261</sup></span> </p> <ul><li>파라콤팩트 <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/%EA%B5%AD%EC%86%8C_%EA%B1%B0%EB%A6%AC%ED%99%94_%EA%B0%80%EB%8A%A5_%EA%B3%B5%EA%B0%84" class="mw-redirect" title="국소 거리화 가능 공간">국소 거리화 가능 공간</a>이다.</li> <li><a href="/wiki/%EA%B1%B0%EB%A6%AC%ED%99%94_%EA%B0%80%EB%8A%A5_%EA%B3%B5%EA%B0%84" title="거리화 가능 공간">거리화 가능 공간</a>이다.</li></ul> <p>따라서, 파라콤팩트 공간의 개념은 <a href="/wiki/%EA%B1%B0%EB%A6%AC%ED%99%94_%EA%B0%80%EB%8A%A5%EC%84%B1" class="mw-redirect" title="거리화 가능성">거리화 가능성</a>과 밀접하게 연관되어 있다. 특히, 모든 <a href="/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간">거리 공간</a>은 파라콤팩트 공간이다. </p><p>이 밖에도, 파라콤팩트 <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 공간</a>에 대하여 다음이 성립한다. </p> <ul><li><a href="/w/index.php?title=%EC%99%84%EC%A0%84_%EC%82%AC%EC%83%81&action=edit&redlink=1" class="new" title="완전 사상 (없는 문서)">완전 사상</a>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">perfect map</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\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>가 파라콤팩트 공간일 때 <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>도 파라콤팩트 공간이다.<sup id="cite_ref-Munkres_4-7" class="reference"><a href="#cite_note-Munkres-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:260, Exercise 8.(a)</sup></span> <a href="/wiki/%EB%8B%AB%ED%9E%8C_%ED%95%A8%EC%88%98" class="mw-redirect" title="닫힌 함수">닫힌</a> <a href="/wiki/%EC%97%B0%EC%86%8D_%ED%95%A8%EC%88%98" title="연속 함수">연속 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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></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}"> <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 파라콤팩트 <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 공간</a>일 때 <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(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b884e2d65b3356219702968b6751485fb8f38570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.068ex; height:2.843ex;" alt="{\displaystyle f(X)}"></span> 역시 파라콤팩트 <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 공간</a>이다.<sup id="cite_ref-MichaelErnestAnother_10-0" class="reference"><a href="#cite_note-MichaelErnestAnother-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:823, Corollary 1</sup></span><sup id="cite_ref-Willard_5-5" class="reference"><a href="#cite_note-Willard-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:148, Theorem 20.12.(b)</sup></span><sup id="cite_ref-Munkres_4-8" class="reference"><a href="#cite_note-Munkres-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:260, Exercise 8.(b)</sup></span></li> <li><a href="/w/index.php?title=%EC%99%84%EC%A0%84_%EC%82%AC%EC%83%81&action=edit&redlink=1" class="new" title="완전 사상 (없는 문서)">완전 사상</a>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">perfect map</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\colon X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>가 메타콤팩트 <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 공간</a>일 때 <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>도 메타콤팩트 <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 공간</a>이다.<sup id="cite_ref-Engelking_11-0" class="reference"><a href="#cite_note-Engelking-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:329, Exercise 5.3.H</sup></span> <a href="/wiki/%EB%8B%AB%ED%9E%8C_%ED%95%A8%EC%88%98" class="mw-redirect" title="닫힌 함수">닫힌</a> <a href="/wiki/%EC%97%B0%EC%86%8D_%ED%95%A8%EC%88%98" title="연속 함수">연속 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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}"></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}"> <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 메타콤팩트 <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 공간</a>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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>가 <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 공간</a>일 때, <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(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b884e2d65b3356219702968b6751485fb8f38570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.068ex; height:2.843ex;" alt="{\displaystyle f(X)}"></span> 역시 메타콤팩트 <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 공간</a>이다.<sup id="cite_ref-Engelking_11-1" class="reference"><a href="#cite_note-Engelking-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:325, Theorem 5.3.7</sup></span></li> <li><a href="/wiki/%EC%A0%95%EA%B7%9C_%EA%B3%B5%EA%B0%84" title="정규 공간">정규</a> <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 공간</a>의 <a href="/wiki/%EC%9C%A0%ED%95%9C_%EC%A7%91%ED%95%A9" title="유한 집합">유한 개</a> 파라콤팩트 <a href="/wiki/%EB%8B%AB%ED%9E%8C%EC%A7%91%ED%95%A9" class="mw-redirect" title="닫힌집합">닫힌집합</a>들의 <a href="/wiki/%ED%95%A9%EC%A7%91%ED%95%A9" title="합집합">합집합</a> 역시 파라콤팩트 집합이다.<sup id="cite_ref-Munkres_4-9" class="reference"><a href="#cite_note-Munkres-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:260</sup></span></li> <li><a href="/wiki/%EC%A0%95%EA%B7%9C_%EA%B3%B5%EA%B0%84" title="정규 공간">정규</a> <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 공간</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}"> <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> 속의 <a href="/wiki/%EA%B0%80%EC%82%B0_%EC%A7%91%ED%95%A9" title="가산 집합">가산 개</a> 파라콤팩트 <a href="/wiki/%EB%8B%AB%ED%9E%8C%EC%A7%91%ED%95%A9" class="mw-redirect" title="닫힌집합">닫힌집합</a>들의 <a href="/wiki/%EB%82%B4%EB%B6%80_(%EC%9C%84%EC%83%81%EC%88%98%ED%95%99)" title="내부 (위상수학)">내부</a>가 이루는 <a href="/wiki/%EC%A7%91%ED%95%A9%EC%A1%B1" title="집합족">집합족</a>이 <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/%EB%8D%AE%EA%B0%9C_(%EC%9C%84%EC%83%81%EC%88%98%ED%95%99)" title="덮개 (위상수학)">덮개</a>를 이룰 때, 그 합집합 역시 파라콤팩트 집합이다.<sup id="cite_ref-Munkres_4-10" class="reference"><a href="#cite_note-Munkres-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:260</sup></span></li></ul> <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=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84&action=edit&section=7" title="부분 편집: 예"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EA%B8%B4_%EC%A7%81%EC%84%A0" title="긴 직선">긴 직선</a>은 <a href="/wiki/%EA%B5%AD%EC%86%8C_%EC%BD%A4%ED%8C%A9%ED%8A%B8" class="mw-redirect" title="국소 콤팩트">국소 콤팩트</a> <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 공간</a>이지만, 파라콤팩트 공간이 아니다. </p><p><a href="/wiki/%EC%A1%B0%EB%A5%B4%EA%B2%90%ED%94%84%EB%9D%BC%EC%9D%B4_%EC%A7%81%EC%84%A0" class="mw-redirect" title="조르겐프라이 직선">조르겐프라이 직선</a>은 파라콤팩트 공간이지만, 두 조르겐프라이 직선의 <a href="/wiki/%EA%B3%B1%EA%B3%B5%EA%B0%84" class="mw-redirect" title="곱공간">곱공간</a>은 파라콤팩트 공간이 아니다. </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=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84&action=edit&section=8" title="부분 편집: 역사"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>1940년에 존 윌더 튜키(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">John Wilder Tukey</span>)는 "완전 정규 공간"(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">fully normal space</span>)이라는 개념을 정의하였다.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SS_13-0" class="reference"><a href="#cite_note-SS-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:165</sup></span> 1944년에 <a href="/wiki/%ED%94%84%EB%9E%91%EC%8A%A4" title="프랑스">프랑스</a>의 수학자 <a href="/wiki/%EC%9E%A5_%EB%94%94%EC%99%B8%EB%8F%84%EB%84%A4" title="장 디외도네">장 디외도네</a>는 파라콤팩트 공간의 개념을 정의하였다.<sup id="cite_ref-SS_13-1" class="reference"><a href="#cite_note-SS-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:165</sup></span><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> 1948년에 아서 해럴드 스톤(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Arthur Harold Stone</span>)은 완전 정규 공간의 개념과 파라콤팩트 공간의 개념이 (<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 조건</a> 아래) 서로 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>임을 증명하였다.<sup id="cite_ref-SS_13-2" class="reference"><a href="#cite_note-SS-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:165</sup></span><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/%EB%AA%A8%EB%A6%AC%ED%83%80_%EC%A0%95%EB%A6%AC" class="mw-redirect" title="모리타 정리">모리타 정리</a>는 <a href="/wiki/%EB%AA%A8%EB%A6%AC%ED%83%80_%EA%B8%B0%EC%9D%B4%EC%B9%98" title="모리타 기이치">모리타 기이치</a>가 1948년에 증명하였다.<sup id="cite_ref-Morita_7-1" class="reference"><a href="#cite_note-Morita-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SS_13-3" class="reference"><a href="#cite_note-SS-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:165</sup></span> </p> <div class="mw-heading mw-heading2"><h2 id="참고_문헌"><span id=".EC.B0.B8.EA.B3.A0_.EB.AC.B8.ED.97.8C"></span>참고 문헌</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84&action=edit&section=9" title="부분 편집: 참고 문헌"><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 mw-references-columns"><ol class="references"> <li id="cite_note-조용승-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-조용승_1-0">가</a></sup> <sup><a href="#cite_ref-조용승_1-1">나</a></sup></span> <span class="reference-text"><cite class="citation book">조용승 (2010). 《위상수학》. 경문사.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%EC%9C%84%EC%83%81%EC%88%98%ED%95%99&rft.pub=%EA%B2%BD%EB%AC%B8%EC%82%AC&rft.date=2010&rft.au=%EC%A1%B0%EC%9A%A9%EC%8A%B9&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" 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">Pearl, Elliott (2007). 《Open Problems in Topology II》 (영어). Elsevier. <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%B0/0-444-52208-5" title="특수:책찾기/0-444-52208-5"><bdi>0-444-52208-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Open+Problems+in+Topology+II&rft.pub=Elsevier&rft.date=2007&rft.isbn=0-444-52208-5&rft.aulast=Pearl&rft.aufirst=Elliott&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" 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">Hart, K.P.; Nagata, J.; Vaughan, J.E. (2004). 《Encyclopedia of General Topology》 (영어). Elsevier. <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%B0/0-444-50355-2" title="특수:책찾기/0-444-50355-2"><bdi>0-444-50355-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopedia+of+General+Topology&rft.pub=Elsevier&rft.date=2004&rft.isbn=0-444-50355-2&rft.aulast=Hart&rft.aufirst=K.P.&rft.au=Nagata%2C+J.&rft.au=Vaughan%2C+J.E.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Munkres-4"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Munkres_4-0">가</a></sup> <sup><a href="#cite_ref-Munkres_4-1">나</a></sup> <sup><a href="#cite_ref-Munkres_4-2">다</a></sup> <sup><a href="#cite_ref-Munkres_4-3">라</a></sup> <sup><a href="#cite_ref-Munkres_4-4">마</a></sup> <sup><a href="#cite_ref-Munkres_4-5">바</a></sup> <sup><a href="#cite_ref-Munkres_4-6">사</a></sup> <sup><a href="#cite_ref-Munkres_4-7">아</a></sup> <sup><a href="#cite_ref-Munkres_4-8">자</a></sup> <sup><a href="#cite_ref-Munkres_4-9">차</a></sup> <sup><a href="#cite_ref-Munkres_4-10">카</a></sup></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/%EC%A0%9C%EC%9E%84%EC%8A%A4_%EB%A9%8D%ED%81%AC%EB%A0%88%EC%8A%A4" title="제임스 멍크레스">Munkres, James R.</a> (2000). <a rel="nofollow" class="external text" href="http://www.pearsonhighered.com/bookseller/product/Topology/9780131816299.page">《Topology》</a> (영어) 2판. Prentice Hall. <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%B0/978-0-13-181629-9" title="특수:책찾기/978-0-13-181629-9"><bdi>978-0-13-181629-9</bdi></a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0464128">0464128</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:0951.54001">0951.54001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology&rft.edition=2&rft.pub=Prentice+Hall&rft.date=2000&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0951.54001&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0464128&rft.isbn=978-0-13-181629-9&rft.aulast=Munkres&rft.aufirst=James+R.&rft_id=http%3A%2F%2Fwww.pearsonhighered.com%2Fbookseller%2Fproduct%2FTopology%2F9780131816299.page&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Willard-5"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Willard_5-0">가</a></sup> <sup><a href="#cite_ref-Willard_5-1">나</a></sup> <sup><a href="#cite_ref-Willard_5-2">다</a></sup> <sup><a href="#cite_ref-Willard_5-3">라</a></sup> <sup><a href="#cite_ref-Willard_5-4">마</a></sup> <sup><a href="#cite_ref-Willard_5-5">바</a></sup></span> <span class="reference-text"><cite class="citation book">Willard, Stephen (1970). 《General topology》. Addison-Wesley Series in Mathematics (영어). Reading, Massachusetts; Menlo Park, California; London; Don Mills, Ontario: Addison-Wesley. <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%B0/978-0-201-08707-9" title="특수:책찾기/978-0-201-08707-9"><bdi>978-0-201-08707-9</bdi></a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0264581">0264581</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:0205.26601">0205.26601</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+topology&rft.place=Reading%2C+Massachusetts%3B+Menlo+Park%2C+California%3B+London%3B+Don+Mills%2C+Ontario&rft.series=Addison-Wesley+Series+in+Mathematics&rft.pub=Addison-Wesley&rft.date=1970&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0205.26601&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0264581&rft.isbn=978-0-201-08707-9&rft.aulast=Willard&rft.aufirst=Stephen&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">B.M. Scott, "Towards a product theory for orthocompactness", <i>Studies in Topology</i>, N.M. Stavrakas and K.R. Allen, eds (1975), p.517–537.</span> </li> <li id="cite_note-Morita-7"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Morita_7-0">가</a></sup> <sup><a href="#cite_ref-Morita_7-1">나</a></sup></span> <span class="reference-text"><cite class="citation journal"><a href="/wiki/%EB%AA%A8%EB%A6%AC%ED%83%80_%EA%B8%B0%EC%9D%B4%EC%B9%98" title="모리타 기이치">Morita, Kiiti</a> (1948). “Star-finite coverings and the star-finite property”. 《Mathematica Japonicae》 (영어) <b>1</b>: 60-68. <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:0041.09704">0041.09704</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematica+Japonicae&rft.atitle=Star-finite+coverings+and+the++star-finite+property&rft.volume=1&rft.pages=60-68&rft.date=1948&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0041.09704&rft.aulast=Morita&rft.aufirst=Kiiti&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Künzi-8"><span class="mw-cite-backlink"><a href="#cite_ref-Künzi_8-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Künzi, Hans-Peter A. (1987). “Kelley’s conjecture and preorthocompactness”. 《Topology and its Applications》 (영어) <b>26</b> (1): 13–23. <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%2F0166-8641%2887%2990022-8">10.1016/0166-8641(87)90022-8</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/0166-8641">0166-8641</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0893800">0893800</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:0623.54012">0623.54012</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Topology+and+its+Applications&rft.atitle=Kelley%E2%80%99s+conjecture+and+preorthocompactness&rft.volume=26&rft.issue=1&rft.pages=13-23&rft.date=1987&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0623.54012&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0893800&rft.issn=0166-8641&rft_id=info%3Adoi%2F10.1016%2F0166-8641%2887%2990022-8&rft.aulast=K%C3%BCnzi&rft.aufirst=Hans-Peter+A.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Gulden-9"><span class="mw-cite-backlink"><a href="#cite_ref-Gulden_9-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Gulden, S. L.; Fleischman, W. M. (1970). “Linearly ordered topological spaces”. 《Proceedings of the American Mathematical Society》 (영어) <b>24</b>: 197–203. <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/10.2307%2F2036727">10.2307/2036727</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/0002-9939">0002-9939</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0250272">0250272</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:0203.55104">0203.55104</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=Linearly+ordered+topological+spaces&rft.volume=24&rft.pages=197-203&rft.date=1970&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0203.55104&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0250272&rft.issn=0002-9939&rft_id=info%3Adoi%2F10.2307%2F2036727&rft.aulast=Gulden&rft.aufirst=S.+L.&rft.au=Fleischman%2C+W.+M.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-MichaelErnestAnother-10"><span class="mw-cite-backlink"><a href="#cite_ref-MichaelErnestAnother_10-0">↑</a></span> <span class="reference-text"><cite class="citation journal">Michael, Ernest (1957). “Another note on paracompact spaces”. 《Proceedings of the American Mathematical Society》 (영어) <b>8</b>: 822–828. <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/10.2307%2F2033306">10.2307/2033306</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/0002-9939">0002-9939</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0087079">0087079</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:0078.14805">0078.14805</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+American+Mathematical+Society&rft.atitle=Another+note+on+paracompact+spaces&rft.volume=8&rft.pages=822-828&rft.date=1957&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0078.14805&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0087079&rft.issn=0002-9939&rft_id=info%3Adoi%2F10.2307%2F2033306&rft.aulast=Michael&rft.aufirst=Ernest&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Engelking-11"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Engelking_11-0">가</a></sup> <sup><a href="#cite_ref-Engelking_11-1">나</a></sup></span> <span class="reference-text"><cite class="citation book">Engelking, Ryszard (1989). 《General topology》. Sigma Series in Pure Mathematics (영어) <b>6</b> 개정 완결판. Berlin: Heldermann Verlag. <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%B0/3-88538-006-4" title="특수:책찾기/3-88538-006-4"><bdi>3-88538-006-4</bdi></a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=1039321">1039321</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:0684.54001">0684.54001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+topology&rft.place=Berlin&rft.series=Sigma+Series+in+Pure+Mathematics&rft.edition=%EA%B0%9C%EC%A0%95+%EC%99%84%EA%B2%B0&rft.pub=Heldermann+Verlag&rft.date=1989&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0684.54001&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1039321&rft.isbn=3-88538-006-4&rft.aulast=Engelking&rft.aufirst=Ryszard&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a></span> <span class="reference-text"><cite class="citation journal">Tukey, John W. (1940). “Convergence and Uniformity in Topology”. Annals of Mathematics Studies (영어) <b>2</b>. Princeton University Press. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0002515">0002515</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Convergence+and+Uniformity+in+Topology&rft.volume=2&rft.date=1940&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0002515&rft.aulast=Tukey&rft.aufirst=John+W.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-SS-13"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-SS_13-0">가</a></sup> <sup><a href="#cite_ref-SS_13-1">나</a></sup> <sup><a href="#cite_ref-SS_13-2">다</a></sup> <sup><a href="#cite_ref-SS_13-3">라</a></sup></span> <span class="reference-text"><cite class="citation book">Steen, Lynn Arthur; Seebach, J. Arthur, Jr. (1978). 《Counterexamples in topology》 (영어) 2판. Springer. <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%2F978-1-4612-6290-9">10.1007/978-1-4612-6290-9</a>. <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%B0/978-0-387-90312-5" title="특수:책찾기/978-0-387-90312-5"><bdi>978-0-387-90312-5</bdi></a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=507446">507446</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:0386.54001">0386.54001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Counterexamples+in+topology&rft.edition=2&rft.pub=Springer&rft.date=1978&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0386.54001&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D507446&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-6290-9&rft.isbn=978-0-387-90312-5&rft.aulast=Steen&rft.aufirst=Lynn+Arthur&rft.au=Seebach%2C+J.+Arthur%2C+Jr.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="#cite_ref-14">↑</a></span> <span class="reference-text"><cite class="citation journal"><a href="/wiki/%EC%9E%A5_%EB%94%94%EC%99%B8%EB%8F%84%EB%84%A4" title="장 디외도네">Dieudonné, Jean</a> (1944). “Une généralisation des espaces compacts”. 《Journal de mathématiques pures et appliquées (neuvième série)》 (프랑스어) <b>23</b>: 65–76. <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/0021-7824">0021-7824</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0013297">0013297</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+de+math%C3%A9matiques+pures+et+appliqu%C3%A9es+%28neuvi%C3%A8me+s%C3%A9rie%29&rft.atitle=Une+g%C3%A9n%C3%A9ralisation+des+espaces+compacts&rft.volume=23&rft.pages=65-76&rft.date=1944&rft.issn=0021-7824&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0013297&rft.aulast=Dieudonn%C3%A9&rft.aufirst=Jean&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><a href="#cite_ref-15">↑</a></span> <span class="reference-text"><cite class="citation journal">Stone, A. H. (1948년 10월). “Paracompactness and product spaces” (영어) <b>54</b> (10). <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/10.1090%2FS0002-9904-1948-09118-2">10.1090/S0002-9904-1948-09118-2</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/0273-0979">0273-0979</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0026802">0026802</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:0032.31403">0032.31403</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Paracompactness+and+product+spaces&rft.volume=54&rft.issue=10&rft.date=1948-10&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0032.31403&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0026802&rft.issn=0273-0979&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1948-09118-2&rft.aulast=Stone&rft.aufirst=A.+H.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <ul><li><cite class="citation book">Fletcher, P.; Lindgren, W. F. (1982). 《Quasi-uniform spaces》 (영어). Marcel Dekker. <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%B0/0-8247-1839-9" title="특수:책찾기/0-8247-1839-9"><bdi>0-8247-1839-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quasi-uniform+spaces&rft.pub=Marcel+Dekker&rft.date=1982&rft.isbn=0-8247-1839-9&rft.aulast=Fletcher&rft.aufirst=P.&rft.au=Lindgren%2C+W.+F.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></li></ul> <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=%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84&action=edit&section=10" title="부분 편집: 외부 링크"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation web"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Paracompact_space">“Paracompact space”</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=Paracompact+space&rft.date=2001&rft.isbn=978-1-55608-010-4&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FParacompact_space&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" 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/ParacompactSpace.html">“Paracompact space”</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=Paracompact+space&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FParacompactSpace.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" 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/paracompact+topological+space">“Paracompact topological space”</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=Paracompact+topological+space&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Fparacompact%2Btopological%2Bspace&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="http://topospaces.subwiki.org/wiki/Paracompact_space">“Paracompact space”</a>. 《Topospaces》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Topospaces&rft.atitle=Paracompact+space&rft_id=http%3A%2F%2Ftopospaces.subwiki.org%2Fwiki%2FParacompact_space&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://dantopology.wordpress.com/2012/11/08/cartesian-products-of-two-paracompact-spaces/">“Cartesian products of two paracompact spaces”</a>. 《Dan Ma’s Topology Blog》 (영어). 2012년 11월 8일.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Dan+Ma%E2%80%99s+Topology+Blog&rft.atitle=Cartesian+products+of+two+paracompact+spaces&rft.date=2012-11-08&rft_id=https%3A%2F%2Fdantopology.wordpress.com%2F2012%2F11%2F08%2Fcartesian-products-of-two-paracompact-spaces%2F&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://dantopology.wordpress.com/2009/10/18/ccc-paracompact-lindelof/">“CCC + Paracompact => Lindelof”</a>. 《Dan Ma’s Topology Blog》 (영어). 2009년 10월 18일.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Dan+Ma%E2%80%99s+Topology+Blog&rft.atitle=CCC+%2B+Paracompact+%3D%3E+Lindelof&rft.date=2009-10-18&rft_id=https%3A%2F%2Fdantopology.wordpress.com%2F2009%2F10%2F18%2Fccc-paracompact-lindelof%2F&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%8C%8C%EB%9D%BC%EC%BD%A4%ED%8C%A9%ED%8A%B8+%EA%B3%B5%EA%B0%84" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r36480591">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output 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