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href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</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="站点"> <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="目录" 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">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</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">序言</div> </a> </li> <li id="toc-簡說" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#簡說"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>簡說</span> </div> </a> <ul id="toc-簡說-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-定义" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#定义"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>定义</span> </div> </a> <button aria-controls="toc-定义-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>开关定义子章节</span> </button> <ul id="toc-定义-sublist" class="vector-toc-list"> <li id="toc-实区间" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#实区间"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>实区间</span> </div> </a> <ul id="toc-实区间-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-偏序集或预序集中的区间" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#偏序集或预序集中的区间"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>偏序集或预序集中的区间</span> </div> </a> <ul id="toc-偏序集或预序集中的区间-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-序凸集和序凸分支" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#序凸集和序凸分支"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>序凸集和序凸分支</span> </div> </a> <ul id="toc-序凸集和序凸分支-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-區間算術" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#區間算術"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>區間算術</span> </div> </a> <ul id="toc-區間算術-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-另一種寫法" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#另一種寫法"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>另一種寫法</span> </div> </a> <ul id="toc-另一種寫法-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參考" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參考"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>參考</span> </div> </a> <ul id="toc-參考-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="目录" 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="开关目录" > <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">开关目录</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">區間</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="前往另一种语言写成的文章。66种语言可用" > <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-66" 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">66种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AC%D8%A7%D9%84_%D9%81%D8%A7%D8%B5%D9%84_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="مجال فاصل (رياضيات) – 阿拉伯语" lang="ar" hreflang="ar" data-title="مجال فاصل (رياضيات)" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C6%8Fd%C9%99di_aral%C4%B1qlar" title="Ədədi aralıqlar – 阿塞拜疆语" lang="az" hreflang="az" data-title="Ədədi aralıqlar" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%B4%D1%86%D1%96%D0%BD%D0%B0%D0%BA" title="Адцінак – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Адцінак" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D1%80%D0%B2%D0%B0%D0%BB_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Интервал (математика) – 保加利亚语" lang="bg" hreflang="bg" data-title="Интервал (математика)" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Interval (matematika)" data-language-autonym="Bosanski" data-language-local-name="波斯尼亚语" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Interval_(matem%C3%A0tiques)" title="Interval (matemàtiques) – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Interval (matemàtiques)" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%D8%A7%D9%88%DB%95_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="ماوە (ماتماتیک) – 中库尔德语" lang="ckb" hreflang="ckb" data-title="ماوە (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="中库尔德语" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – 捷克语" lang="cs" hreflang="cs" data-title="Interval (matematika)" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%D1%83%D1%88%C4%83%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Хушăк (математика) – 楚瓦什语" lang="cv" hreflang="cv" data-title="Хушăк (математика)" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什语" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Cyfwng" title="Cyfwng – 威尔士语" lang="cy" hreflang="cy" data-title="Cyfwng" data-language-autonym="Cymraeg" data-language-local-name="威尔士语" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Interval_(matematik)" title="Interval (matematik) – 丹麦语" lang="da" hreflang="da" data-title="Interval (matematik)" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Intervall_(Mathematik)" title="Intervall (Mathematik) – 德语" lang="de" hreflang="de" data-title="Intervall (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%AC%CF%83%CF%84%CE%B7%CE%BC%CE%B1_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Διάστημα (μαθηματικά) – 希腊语" lang="el" hreflang="el" data-title="Διάστημα (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Interval_(mathematics)" title="Interval (mathematics) – 英语" lang="en" hreflang="en" data-title="Interval (mathematics)" data-language-autonym="English" data-language-local-name="英语" 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/Intervalo_(matematiko)" title="Intervalo (matematiko) – 世界语" lang="eo" hreflang="eo" data-title="Intervalo (matematiko)" data-language-autonym="Esperanto" data-language-local-name="世界语" 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/Intervalo_(matem%C3%A1tica)" title="Intervalo (matemática) – 西班牙语" lang="es" hreflang="es" data-title="Intervalo (matemática)" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Intervall_(matemaatika)" title="Intervall (matemaatika) – 爱沙尼亚语" lang="et" hreflang="et" data-title="Intervall (matemaatika)" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Tarte_(matematika)" title="Tarte (matematika) – 巴斯克语" lang="eu" hreflang="eu" data-title="Tarte (matematika)" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A8%D8%A7%D8%B2%D9%87" title="بازه – 波斯语" lang="fa" hreflang="fa" data-title="بازه" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/V%C3%A4li" title="Väli – 芬兰语" lang="fi" hreflang="fi" data-title="Väli" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Intervalle_(math%C3%A9matiques)" title="Intervalle (mathématiques) – 法语" lang="fr" hreflang="fr" data-title="Intervalle (mathématiques)" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Interval_(matematiche)" title="Interval (matematiche) – 弗留利语" lang="fur" hreflang="fur" data-title="Interval (matematiche)" data-language-autonym="Furlan" data-language-local-name="弗留利语" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Eatramh_(matamaitic)" title="Eatramh (matamaitic) – 爱尔兰语" lang="ga" hreflang="ga" data-title="Eatramh (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="爱尔兰语" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Intervalo_(matem%C3%A1ticas)" title="Intervalo (matemáticas) – 加利西亚语" lang="gl" hreflang="gl" data-title="Intervalo (matemáticas)" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%98%D7%A2_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="קטע (מתמטיקה) – 希伯来语" lang="he" hreflang="he" data-title="קטע (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%A4%E0%A4%B0%E0%A4%BE%E0%A4%B2_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="अंतराल (गणित) – 印地语" lang="hi" hreflang="hi" data-title="अंतराल (गणित)" data-language-autonym="हिन्दी" data-language-local-name="印地语" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – 克罗地亚语" lang="hr" hreflang="hr" data-title="Interval (matematika)" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Intervallum" title="Intervallum – 匈牙利语" lang="hu" hreflang="hu" data-title="Intervallum" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%AB%D5%BB%D5%A1%D5%AF%D5%A1%D5%B5%D6%84_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Միջակայք (մաթեմատիկա) – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Միջակայք (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Selang_(matematika)" title="Selang (matematika) – 印度尼西亚语" lang="id" hreflang="id" data-title="Selang (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Intervalo_(matematiko)" title="Intervalo (matematiko) – 伊多语" lang="io" hreflang="io" data-title="Intervalo (matematiko)" data-language-autonym="Ido" data-language-local-name="伊多语" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Bil_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Bil (stærðfræði) – 冰岛语" lang="is" hreflang="is" data-title="Bil (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="冰岛语" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Intervallo_(matematica)" title="Intervallo (matematica) – 意大利语" lang="it" hreflang="it" data-title="Intervallo (matematica)" data-language-autonym="Italiano" data-language-local-name="意大利语" 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/%E5%8C%BA%E9%96%93_(%E6%95%B0%E5%AD%A6)" title="区間 (数学) – 日语" lang="ja" hreflang="ja" data-title="区間 (数学)" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D1%80%D0%B2%D0%B0%D0%BB" title="Интервал – 哈萨克语" lang="kk" hreflang="kk" data-title="Интервал" data-language-autonym="Қазақша" data-language-local-name="哈萨克语" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%AC%EA%B0%84" title="구간 – 韩语" lang="ko" hreflang="ko" data-title="구간" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Intervallum_(mathematica)" title="Intervallum (mathematica) – 拉丁语" lang="la" hreflang="la" data-title="Intervallum (mathematica)" data-language-autonym="Latina" data-language-local-name="拉丁语" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Intervalas_(matematika)" title="Intervalas (matematika) – 立陶宛语" lang="lt" hreflang="lt" data-title="Intervalas (matematika)" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Interv%C4%81ls_(matem%C4%81tika)" title="Intervāls (matemātika) – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Intervāls (matemātika)" data-language-autonym="Latviešu" data-language-local-name="拉脱维亚语" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D1%80%D0%B2%D0%B0%D0%BB_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Интервал (математика) – 马其顿语" lang="mk" hreflang="mk" data-title="Интервал (математика)" data-language-autonym="Македонски" data-language-local-name="马其顿语" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Interval_(wiskunde)" title="Interval (wiskunde) – 荷兰语" lang="nl" hreflang="nl" data-title="Interval (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Intervall_i_matematikk" title="Intervall i matematikk – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Intervall i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Intervall_(matematikk)" title="Intervall (matematikk) – 书面挪威语" lang="nb" hreflang="nb" data-title="Intervall (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przedzia%C5%82_(matematyka)" title="Przedział (matematyka) – 波兰语" lang="pl" hreflang="pl" data-title="Przedział (matematyka)" data-language-autonym="Polski" data-language-local-name="波兰语" 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/Intervalo_(matem%C3%A1tica)" title="Intervalo (matemática) – 葡萄牙语" lang="pt" hreflang="pt" data-title="Intervalo (matemática)" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Interval_(matematic%C4%83)" title="Interval (matematică) – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Interval (matematică)" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%BC%D0%B5%D0%B6%D1%83%D1%82%D0%BE%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Промежуток (математика) – 俄语" lang="ru" hreflang="ru" data-title="Промежуток (математика)" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Interval_(mathematics)" title="Interval (mathematics) – 苏格兰语" lang="sco" hreflang="sco" data-title="Interval (mathematics)" data-language-autonym="Scots" data-language-local-name="苏格兰语" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Interval (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Interval_(mathematics)" title="Interval (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Interval (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Interval (matematika)" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Interval (matematika)" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Intervali_(matematik%C3%AB)" title="Intervali (matematikë) – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Intervali (matematikë)" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D1%80%D0%B2%D0%B0%D0%BB_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Интервал (математика) – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Интервал (математика)" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Interval_(matematika)" title="Interval (matematika) – 巽他语" lang="su" hreflang="su" data-title="Interval (matematika)" data-language-autonym="Sunda" data-language-local-name="巽他语" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Intervall_(matematik)" title="Intervall (matematik) – 瑞典语" lang="sv" hreflang="sv" data-title="Intervall (matematik)" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%9F%E0%AF%88%E0%AE%B5%E0%AF%86%E0%AE%B3%E0%AE%BF_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="இடைவெளி (கணிதம்) – 泰米尔语" lang="ta" hreflang="ta" data-title="இடைவெளி (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%8A%E0%B9%88%E0%B8%A7%E0%B8%87_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="ช่วง (คณิตศาสตร์) – 泰语" lang="th" hreflang="th" data-title="ช่วง (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%BC%D1%96%D0%B6%D0%BE%D0%BA_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Проміжок (математика) – 乌克兰语" lang="uk" hreflang="uk" data-title="Проміжок (математика)" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%88%D9%82%D9%81%DB%81_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="وقفہ (ریاضی) – 乌尔都语" lang="ur" hreflang="ur" data-title="وقفہ (ریاضی)" data-language-autonym="اردو" data-language-local-name="乌尔都语" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Interval" title="Interval – 乌兹别克语" lang="uz" hreflang="uz" data-title="Interval" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="乌兹别克语" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kho%E1%BA%A3ng_(to%C3%A1n_h%E1%BB%8Dc)" title="Khoảng (toán học) – 越南语" lang="vi" hreflang="vi" data-title="Khoảng (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8C%BA%E9%97%B4" title="区间 – 吴语" lang="wuu" hreflang="wuu" data-title="区间" data-language-autonym="吴语" data-language-local-name="吴语" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%97%D0%B0%D0%B2%D1%81%D0%B0%D1%80_(%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Завсар (Математика) – 卡尔梅克语" lang="xal" hreflang="xal" data-title="Завсар (Математика)" data-language-autonym="Хальмг" data-language-local-name="卡尔梅克语" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%8D%80%E9%96%93" title="區間 – 文言文" lang="lzh" hreflang="lzh" data-title="區間" data-language-autonym="文言" data-language-local-name="文言文" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%96%93%E8%B7%9D" title="間距 – 粤语" lang="yue" hreflang="yue" data-title="間距" data-language-autonym="粵語" data-language-local-name="粤语" class="interlanguage-link-target"><span>粵語</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/Q185148#sitelinks-wikipedia" title="编辑跨语言链接" 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"}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable"><span class="noviewer hatnote-notice-img" typeof="mw:File"><a href="/wiki/Wikipedia:%E6%B6%88%E6%AD%A7%E4%B9%89" title="Wikipedia:消歧义"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/25px-Disambig_gray.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/38px-Disambig_gray.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Disambig_gray.svg/50px-Disambig_gray.svg.png 2x" data-file-width="220" data-file-height="168" /></a></span>此條目介紹的是数学上的区间概念。关于铁路运输的区间概念，请见「<b><a href="/wiki/%E9%96%89%E5%A1%9E_(%E9%90%B5%E8%B7%AF)" title="閉塞 (鐵路)">闭塞 (铁路)</a></b>」。</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Interval0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Interval0.png/400px-Interval0.png" decoding="async" width="400" height="79" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/10/Interval0.png 1.5x" data-file-width="419" data-file-height="83" /></a><figcaption>在圖中的數軸上，所有大于<i>x</i>和小于<i>x</i>+<i>a</i>的数组成了一个开区间。</figcaption></figure> <p><b>區間</b>（英語：<span lang="en">interval</span>）在<a href="/wiki/%E6%95%B8%E5%AD%B8" class="mw-redirect" title="數學">數學</a>上是指某個範圍的數的集合，或者更一般地是指某个范围的<a href="/wiki/%E9%A0%90%E5%BA%8F%E9%9B%86" class="mw-redirect" title="預序集">预序集</a>元素的集合，一般以<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>形式表示。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="簡說"><span id=".E7.B0.A1.E8.AA.AA"></span>簡說</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8D%80%E9%96%93&action=edit&section=1" title="编辑章节：簡說"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在<a href="/wiki/%E5%88%9D%E7%AD%89%E4%BB%A3%E6%95%B8" title="初等代數">初等代數</a>，傳統上區間指一個<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集</a>，包含在某兩個特定<a href="/wiki/%E5%AF%A6%E6%95%B8" class="mw-redirect" title="實數">實數</a>之間的所有實數，亦可能包含該兩個實數（或其中之一）。區間表示法是表示一個變數在某個區間內的方式。通用的區間表示法中，圓括號表示<b>排除</b>，方括號表示<b>包括</b>。例如，開區間<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 (10,20)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>10</mn> <mo>,</mo> <mn>20</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (10,20)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efca86726b8c693eef643e451ef58447324491dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.493ex; height:2.843ex;" alt="{\displaystyle (10,20)}"></span>表示所有在<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 10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec811eb07dcac7ea67b413c5665390a1671ecb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 10}"></span>和<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 20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a54c80a7183ec4efa84bba969ef7894f5d78e70c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 20}"></span>之間的實數，但不包括<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 10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec811eb07dcac7ea67b413c5665390a1671ecb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 10}"></span>或<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 20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a54c80a7183ec4efa84bba969ef7894f5d78e70c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 20}"></span>。另一方面，閉區間<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 [10,20]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>10</mn> <mo>,</mo> <mn>20</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [10,20]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f50f2201b22faca4cbd340a04f620d1672b4bce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.977ex; height:2.843ex;" alt="{\displaystyle [10,20]}"></span>表示所有在<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 10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec811eb07dcac7ea67b413c5665390a1671ecb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 10}"></span>和<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 20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a54c80a7183ec4efa84bba969ef7894f5d78e70c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 20}"></span>之間的實數，以及<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 10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec811eb07dcac7ea67b413c5665390a1671ecb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 10}"></span>和<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 20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a54c80a7183ec4efa84bba969ef7894f5d78e70c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 20}"></span>。<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="定义"><span id=".E5.AE.9A.E4.B9.89"></span>定义</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8D%80%E9%96%93&action=edit&section=2" title="编辑章节：定义"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="实区间"><span id=".E5.AE.9E.E5.8C.BA.E9.97.B4"></span>实区间</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8D%80%E9%96%93&action=edit&section=3" title="编辑章节：实区间"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在赋予通常序的实数集<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>里，以<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bf0e2324bac843fab916c46d1b3349017a616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {R} }"></span>为端点的<b>开区间</b>和<b>闭区间</b>分别是： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)=\{x\in \mathbb {R} \colon a<x<b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)=\{x\in \mathbb {R} \colon a<x<b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ede13556abc7380ae7ff793caf596757bb1dea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.13ex; height:2.843ex;" alt="{\displaystyle (a,b)=\{x\in \mathbb {R} \colon a<x<b\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]=\{x\in \mathbb {R} \colon a\leq x\leq b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]=\{x\in \mathbb {R} \colon a\leq x\leq b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8c383da36732b5465a962646777e1df581e5d52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.615ex; height:2.843ex;" alt="{\displaystyle [a,b]=\{x\in \mathbb {R} \colon a\leq x\leq b\}}"></span></dd></dl> <p>类似地，以<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span>为端点的两个<b>半开区间</b>定义为： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b]=\{x\in \mathbb {R} \colon a<x\leq b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo>≤</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b]=\{x\in \mathbb {R} \colon a<x\leq b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ac6a7d10410e78402279912c4472921cad0be64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.872ex; height:2.843ex;" alt="{\displaystyle (a,b]=\{x\in \mathbb {R} \colon a<x\leq b\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b)=\{x\in \mathbb {R} \colon a\leq x<b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo><</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b)=\{x\in \mathbb {R} \colon a\leq x<b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8a7f6fb708bc50cb947d159c0726ae52d76927" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.872ex; height:2.843ex;" alt="{\displaystyle [a,b)=\{x\in \mathbb {R} \colon a\leq x<b\}}"></span></dd></dl> <p>在一些上下文中，两个端点要求满足<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span>。这排除了<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}"></span>从而区间或是<a href="/wiki/%E5%8D%95%E5%85%83%E7%B4%A0%E9%9B%86%E5%90%88" title="单元素集合">单元素集合</a>或是<a href="/wiki/%E7%A9%BA%E9%9B%86" title="空集">空集</a>的情形，也排除了<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fc0063781fb9bf4ec7608b2fd11ed6d5b05a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a>b}"></span>从而区间为空集的情形。 </p><p>只有左端点<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>的<b>开区间</b>和<b>半开区间</b>分别如下。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,\infty )=\{x\in \mathbb {R} \colon x>a\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mi>x</mi> <mo>></mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,\infty )=\{x\in \mathbb {R} \colon x>a\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1550169e2f39f579621544f6926ca13419de0ee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.007ex; height:2.843ex;" alt="{\displaystyle (a,\infty )=\{x\in \mathbb {R} \colon x>a\},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,\infty )=\{x\in \mathbb {R} \colon x\geq a\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mi>x</mi> <mo>≥</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,\infty )=\{x\in \mathbb {R} \colon x\geq a\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027c8e66bc7846b296c5fdd7d9037c1a8a54d243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.75ex; height:2.843ex;" alt="{\displaystyle [a,\infty )=\{x\in \mathbb {R} \colon x\geq a\},}"></span></dd></dl> <p>只有右端点<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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>的<b>开区间</b>和<b>半开区间</b>分别如下。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,b)=\{x\in \mathbb {R} \colon x<b\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mi>x</mi> <mo><</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,b)=\{x\in \mathbb {R} \colon x<b\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f578716bc168e4f22ac03355d2590dd649691d7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.351ex; height:2.843ex;" alt="{\displaystyle (-\infty ,b)=\{x\in \mathbb {R} \colon x<b\},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,b]=\{x\in \mathbb {R} \colon x\leq b\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>:</mo> <mi>x</mi> <mo>≤</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,b]=\{x\in \mathbb {R} \colon x\leq b\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4e7ab6336f5e7616cdd7218d3aeace33a41cc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.093ex; height:2.843ex;" alt="{\displaystyle (-\infty ,b]=\{x\in \mathbb {R} \colon x\leq b\},}"></span></dd></dl> <p>整个实数线等于没有端点的区间： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,\infty )=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,\infty )=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3d9463863b5eb37d7b74ae2f81e815fc2bd6d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.075ex; height:2.843ex;" alt="{\displaystyle (-\infty ,\infty )=\mathbb {R} }"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="偏序集或预序集中的区间"><span id=".E5.81.8F.E5.BA.8F.E9.9B.86.E6.88.96.E9.A2.84.E5.BA.8F.E9.9B.86.E4.B8.AD.E7.9A.84.E5.8C.BA.E9.97.B4"></span>偏序集或预序集中的区间</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8D%80%E9%96%93&action=edit&section=4" title="编辑章节：偏序集或预序集中的区间"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>区间的概念在任何<a href="/wiki/%E5%81%8F%E5%BA%8F%E9%9B%86" class="mw-redirect" title="偏序集">偏序集</a>或者更一般地，在任何<a href="/wiki/%E9%A0%90%E5%BA%8F%E9%9B%86" class="mw-redirect" title="預序集">预序集</a>中有定义。对于<a href="/wiki/%E9%A0%90%E5%BA%8F%E9%9B%86" class="mw-redirect" title="預序集">预序集</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\lesssim )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≲</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\lesssim )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0cefd1633a866ad968e394c3b858334f171f0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\lesssim )}"></span>和两个元素<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in X,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32b511d83a7e02a312e4d88755e8929fca8f8b6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.729ex; height:2.509ex;" alt="{\displaystyle a,b\in X,}"></span>，我们可以类似定义<sup id="cite_ref-Vind_2-0" class="reference"><a href="#cite_note-Vind-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:11, Definition 11</sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)=\{x\in X\colon a<x<b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)=\{x\in X\colon a<x<b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43fc8c916d0fc77beb73fd926ee4bb4fff88878a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.432ex; height:2.843ex;" alt="{\displaystyle (a,b)=\{x\in X\colon a<x<b\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]=\{x\in X\colon a\lesssim x\lesssim b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>a</mi> <mo>≲</mo> <mi>x</mi> <mo>≲</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]=\{x\in X\colon a\lesssim x\lesssim b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf1b91d4e0efd987fb45e4de370662d066493f63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.917ex; height:2.843ex;" alt="{\displaystyle [a,b]=\{x\in X\colon a\lesssim x\lesssim b\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b]=\{x\in X\colon a<x\lesssim b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo>≲</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b]=\{x\in X\colon a<x\lesssim b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c65294a9d7cd6a779496abf27cf02d0d1c1b5668" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.174ex; height:2.843ex;" alt="{\displaystyle (a,b]=\{x\in X\colon a<x\lesssim b\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b)=\{x\in X\colon a\lesssim x<b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>a</mi> <mo>≲</mo> <mi>x</mi> <mo><</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b)=\{x\in X\colon a\lesssim x<b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70fe2dadf1e99e126dcd1e55265dbb26e371ecfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.174ex; height:2.843ex;" alt="{\displaystyle [a,b)=\{x\in X\colon a\lesssim x<b\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,\infty )=\{x\in X\colon a<x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,\infty )=\{x\in X\colon a<x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e363dd91c26650c62f7244bc2a3fe545757e14e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.662ex; height:2.843ex;" alt="{\displaystyle (a,\infty )=\{x\in X\colon a<x\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,\infty )=\{x\in X\colon a\lesssim x\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>a</mi> <mo>≲</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,\infty )=\{x\in X\colon a\lesssim x\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21aa08dcd54d7debf708894c16f3a7d1983f7998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.405ex; height:2.843ex;" alt="{\displaystyle [a,\infty )=\{x\in X\colon a\lesssim x\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,b)=\{x\in X\colon x<b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>x</mi> <mo><</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,b)=\{x\in X\colon x<b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992a1dc19b4fd63752ec0db4de10e13e2d636009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.006ex; height:2.843ex;" alt="{\displaystyle (-\infty ,b)=\{x\in X\colon x<b\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,b]=\{x\in X\colon x\lesssim b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>x</mi> <mo>≲</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,b]=\{x\in X\colon x\lesssim b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aac1feb3c6cb6fba7d13cdd4faadf6cc95f62b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.748ex; height:2.843ex;" alt="{\displaystyle (-\infty ,b]=\{x\in X\colon x\lesssim b\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,\infty )=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,\infty )=X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d506ef89d04b1a1b10df4a5570be99b682a7f70c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.377ex; height:2.843ex;" alt="{\displaystyle (-\infty ,\infty )=X}"></span></dd></dl> <p>其中<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x<y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb239de6fee56ea8b6a65f7858d95b87632069f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x<y}"></span>意思是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\lesssim y\not \lesssim x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≲</mo> <mi>y</mi> <mo>≴</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\lesssim y\not \lesssim x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e104152633a40c1e6f7a9c1d0b8b0e68bd7f4fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.013ex; height:2.843ex;" alt="{\displaystyle x\lesssim y\not \lesssim x}"></span>。其实，只有一个端点或者没有端点的区间等同于更大的预序集 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}=X\sqcup \{-\infty ,\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>X</mi> <mo>⊔</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}=X\sqcup \{-\infty ,\infty \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b59614cc64fae001fa1ed13c404627563c5d2879" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.456ex; height:3.009ex;" alt="{\displaystyle {\bar {X}}=X\sqcup \{-\infty ,\infty \}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty <x<\infty \qquad (\forall x\in X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty <x<\infty \qquad (\forall x\in X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ed45a50c51386e2e1e75d357af4f17cd4901ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.88ex; height:2.843ex;" alt="{\displaystyle -\infty <x<\infty \qquad (\forall x\in X)}"></span></dd></dl> <p>上具有两个端点的区间，使得它是<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>的子集。当<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=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2e2b6427cd2b517be352b378a1830c1540e3a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.757ex; height:2.176ex;" alt="{\displaystyle X=\mathbb {R} }"></span>时，可以取<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\mathbb {R} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbb {R} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55c15a406946074613e09e4ae53bdfec849ec523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.509ex;" alt="{\displaystyle {\bar {\mathbb {R} }}}"></span>为<a href="/wiki/%E6%93%B4%E5%B1%95%E5%AF%A6%E6%95%B8%E7%B7%9A" title="擴展實數線">扩展实数线</a>。 </p> <div class="mw-heading mw-heading3"><h3 id="序凸集和序凸分支"><span id=".E5.BA.8F.E5.87.B8.E9.9B.86.E5.92.8C.E5.BA.8F.E5.87.B8.E5.88.86.E6.94.AF"></span>序凸集和序凸分支</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8D%80%E9%96%93&action=edit&section=5" title="编辑章节：序凸集和序凸分支"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E9%A0%90%E5%BA%8F%E9%9B%86" class="mw-redirect" title="預序集">预序集</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\lesssim )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≲</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\lesssim )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0cefd1633a866ad968e394c3b858334f171f0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\lesssim )}"></span>的子集<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dce86da0107830a9a97287f9486d9b4ff022875" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.822ex; height:2.343ex;" alt="{\displaystyle A\subseteq X}"></span>是<b>序凸集</b>，如果对于任意<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd3c2bd17578ba23f55a299668cd7accc5c7a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.103ex; height:2.509ex;" alt="{\displaystyle x,y\in A}"></span>以及任意<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\lesssim z\lesssim y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≲</mo> <mi>z</mi> <mo>≲</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\lesssim z\lesssim y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52c676913d2256a6bb4f87bf6b3ebbe1fda31ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.77ex; height:2.843ex;" alt="{\displaystyle x\lesssim z\lesssim y}"></span>有<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 z\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40028fe140831e54037c7b9f8d6907fdeaf4c256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.672ex; height:2.176ex;" alt="{\displaystyle z\in A}"></span>。与实区间的情形不同，预序集的序凸集不一定是区间。例如，在<a href="/wiki/%E6%9C%89%E7%90%86%E6%95%B0" title="有理数">有理数</a>的<a href="/wiki/%E5%85%A8%E5%BA%8F%E9%9B%86" class="mw-redirect" title="全序集">全序集</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Q} ,\leq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> <mo>≤</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Q} ,\leq )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2268caf1bce76cac74c75d52efe50778718b6c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Q} ,\leq )}"></span>中， </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} =\{x\in \mathbb {Q} \colon x^{2}<2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>:</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} =\{x\in \mathbb {Q} \colon x^{2}<2\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565035171801c8c89e9a8fd52b1d31c88aebd0c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.889ex; height:3.176ex;" alt="{\displaystyle \mathbb {Q} =\{x\in \mathbb {Q} \colon x^{2}<2\}}"></span></dd></dl> <p>是序凸集，但它不是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>的区间，这是因为2的平方根在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>中是不存在的。 </p><p>设<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\lesssim )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>≲</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\lesssim )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0cefd1633a866ad968e394c3b858334f171f0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.631ex; height:2.843ex;" alt="{\displaystyle (X,\lesssim )}"></span>是一个<a href="/wiki/%E9%A0%90%E5%BA%8F%E9%9B%86" class="mw-redirect" title="預序集">预序集</a>，且<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\subseteq X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>⊆</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\subseteq X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abf3e597919d6ae699180ce66b51edbe14f3546a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.852ex; height:2.343ex;" alt="{\displaystyle Y\subseteq X}"></span>。包含在<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>中的<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>的序凸集关于包含关系构成<a href="/wiki/%E5%81%8F%E5%BA%8F%E9%9B%86" class="mw-redirect" title="偏序集">偏序集</a>。这个偏序集的<a href="/wiki/%E6%9E%81%E5%A4%A7%E5%85%83" class="mw-redirect" title="极大元">极大元</a>叫做<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>的<b>序凸分支</b>。<sup id="cite_ref-Heath_3-0" class="reference"><a href="#cite_note-Heath-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference" style="white-space:nowrap;">:Definition 5.1</sup>由<a href="/wiki/%E4%BD%90%E6%81%A9%E5%BC%95%E7%90%86" title="佐恩引理">佐恩引理</a>，包含在<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>中的<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>的任意序凸集包含于<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>的一个序凸分支，然而这种序凸分支不一定是唯一的。在<a href="/wiki/%E5%85%A8%E5%BA%8F%E9%9B%86" class="mw-redirect" title="全序集">全序集</a>中，这样的序凸分支确实唯一。也就是说，<a href="/wiki/%E5%85%A8%E5%BA%8F%E9%9B%86" class="mw-redirect" title="全序集">全序集</a>的子集的序凸分支构成<a href="/wiki/%E9%9B%86%E5%90%88%E5%88%92%E5%88%86" title="集合划分">分划</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="區間算術"><span id=".E5.8D.80.E9.96.93.E7.AE.97.E8.A1.93"></span>區間算術</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8D%80%E9%96%93&action=edit&section=6" title="编辑章节：區間算術"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>區間算術又稱區間數學、區間分析、區間計算，在1950、60年代引進以作數值分析上計算捨去誤差的工具。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\times S=\{x\mid {}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>×</mo> <mi>S</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\times S=\{x\mid {}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44ba39155e6b18a80dd523a4546b37219e2fb7a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.504ex; height:2.843ex;" alt="{\displaystyle T\times S=\{x\mid {}}"></span>屬於<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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>的某些<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/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>，及屬於<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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>的某些<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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>，使得<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y\times z\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>×</mo> <mi>z</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y\times z\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11162d5ed20a5e35c46f2e59b2050bd289ab2c2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.675ex; height:2.843ex;" alt="{\displaystyle x=y\times z\}}"></span></dd></dl> <p>區間算術的基本運算是，對於實數線上的子集<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span>及<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [c,d]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [c,d]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d85b3b21d6d891d97f85e263d394e3c90287586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.55ex; height:2.843ex;" alt="{\displaystyle [c,d]}"></span>： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]+[c,d]=[a+c,b+d]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]+[c,d]=[a+c,b+d]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fee8c5524fea89636103ea04d2271d03fdb2c888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.503ex; height:2.843ex;" alt="{\displaystyle [a,b]+[c,d]=[a+c,b+d]}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]-[c,d]=[a-d,b-c]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>−</mo> <mo stretchy="false">[</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>−</mo> <mi>d</mi> <mo>,</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]-[c,d]=[a-d,b-c]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d65fdee1ba80f2fa2179a29b5938439a15f9f96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.503ex; height:2.843ex;" alt="{\displaystyle [a,b]-[c,d]=[a-d,b-c]}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]\times [c,d]=[\min\{ac,ad,bc,bd\},\max\{ac,ad,bc,bd\}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mi>c</mi> <mo>,</mo> <mi>a</mi> <mi>d</mi> <mo>,</mo> <mi>b</mi> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mi>d</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mi>c</mi> <mo>,</mo> <mi>a</mi> <mi>d</mi> <mo>,</mo> <mi>b</mi> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mi>d</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]\times [c,d]=[\min\{ac,ad,bc,bd\},\max\{ac,ad,bc,bd\}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b334cf018eb4ccf7dcc69bb1fefa02bf878ad27f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.227ex; height:2.843ex;" alt="{\displaystyle [a,b]\times [c,d]=[\min\{ac,ad,bc,bd\},\max\{ac,ad,bc,bd\}]}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {[a,b]}{[c,d]}}=\left[\min \left\{{\frac {a}{c}},{\frac {a}{d}},{\frac {b}{c}},{\frac {b}{d}}\right\},\max \left\{{\frac {a}{c}},{\frac {a}{d}},{\frac {b}{c}},{\frac {b}{d}}\right\}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mo movablelimits="true" form="prefix">min</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>d</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>d</mi> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mo movablelimits="true" form="prefix">max</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>d</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>d</mi> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {[a,b]}{[c,d]}}=\left[\min \left\{{\frac {a}{c}},{\frac {a}{d}},{\frac {b}{c}},{\frac {b}{d}}\right\},\max \left\{{\frac {a}{c}},{\frac {a}{d}},{\frac {b}{c}},{\frac {b}{d}}\right\}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd76a518c678329260f7626ced401e84cce0feed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.571ex; height:6.509ex;" alt="{\displaystyle {\frac {[a,b]}{[c,d]}}=\left[\min \left\{{\frac {a}{c}},{\frac {a}{d}},{\frac {b}{c}},{\frac {b}{d}}\right\},\max \left\{{\frac {a}{c}},{\frac {a}{d}},{\frac {b}{c}},{\frac {b}{d}}\right\}\right]}"></span></dd></dl> <p>被一個包含零的區間除，在基礎區間算術上無定義。 </p><p>加法和乘法符合<a href="/wiki/%E4%BA%A4%E6%8F%9B%E5%BE%8B" title="交換律">交換律</a>、<a href="/wiki/%E7%B5%90%E5%90%88%E5%BE%8B" class="mw-redirect" title="結合律">結合律</a>和子<a href="/wiki/%E5%88%86%E9%85%8D%E5%BE%8B" title="分配律">分配律</a>：集<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(Y+Z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>+</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(Y+Z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/390ba27374aa539b8f4e0c744f242af6a44d1e48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.083ex; height:2.843ex;" alt="{\displaystyle X(Y+Z)}"></span>是<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 XY+XZ}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>Y</mi> <mo>+</mo> <mi>X</mi> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle XY+XZ}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c8565df88316175fb02cc1dd3c0d4be3b140fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.254ex; height:2.343ex;" alt="{\displaystyle XY+XZ}"></span>的子集。 </p> <div class="mw-heading mw-heading2"><h2 id="另一種寫法"><span id=".E5.8F.A6.E4.B8.80.E7.A8.AE.E5.AF.AB.E6.B3.95"></span>另一種寫法</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8D%80%E9%96%93&action=edit&section=7" title="编辑章节：另一種寫法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在<a href="/wiki/%E6%B3%95%E5%9B%BD" title="法国">法国</a>及其他一些<a href="/wiki/%E6%AC%A7%E6%B4%B2" title="欧洲">欧洲</a>国家，用<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 ][}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ][}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/201900e9aecf839f49577253b07fb8d9fe3e0a64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.294ex; height:2.843ex;" alt="{\displaystyle ][}"></span>代替<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 ()}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ()}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7bc8aa05e1302397bb3e7877e842784991351df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.809ex; height:2.843ex;" alt="{\displaystyle ()}"></span>來表示开区间，例如： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left]a,b\right[=\{x\mid a<x<b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>]</mo> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo>[</mo> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left]a,b\right[=\{x\mid a<x<b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1f5b58075e37eab0c81a7e8974553b5d32ae189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.999ex; height:2.843ex;" alt="{\displaystyle \left]a,b\right[=\{x\mid a<x<b\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[a,b\right]=\{x\mid a\leq x\leq b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[a,b\right]=\{x\mid a\leq x\leq b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3541db101303e3fe316f0cfb6aed0024010374" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.999ex; height:2.843ex;" alt="{\displaystyle \left[a,b\right]=\{x\mid a\leq x\leq b\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[a,b\right[=\{x\mid a\leq x<b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo>[</mo> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo><</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[a,b\right[=\{x\mid a\leq x<b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c41672bcc810d0c6045512c30583238d94ae6c2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.999ex; height:2.843ex;" alt="{\displaystyle \left[a,b\right[=\{x\mid a\leq x<b\}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left]a,b\right]=\{x\mid a<x\leq b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>]</mo> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>a</mi> <mo><</mo> <mi>x</mi> <mo>≤</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left]a,b\right]=\{x\mid a<x\leq b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f86c9f682967e3dd77e6d2e094308771219280a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.999ex; height:2.843ex;" alt="{\displaystyle \left]a,b\right]=\{x\mid a<x\leq b\}}"></span></dd></dl> <p><a href="/wiki/%E5%9C%8B%E9%9A%9B%E6%A8%99%E6%BA%96%E5%8C%96%E7%B5%84%E7%B9%94" title="國際標準化組織">國際標準化組織</a>編制的<a href="/w/index.php?title=ISO_31-11&action=edit&redlink=1" class="new" title="ISO 31-11（页面不存在）">ISO 31-11</a>也允許這種寫法<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>。 </p><p>另外，在<a href="/wiki/%E5%B0%8F%E6%95%B8%E9%BB%9E" title="小數點">小數點</a>以逗號來表示的情況下，為免產生混淆，分隔兩數的逗號要用分號來代替，例如將<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 [1,2.3]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2.3</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1,2.3]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76bfc3f12c80028568fde1b67b29c724d371ccca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.462ex; height:2.843ex;" alt="{\displaystyle [1,2.3]}"></span>寫成<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 [1;2,\!3]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>;</mo> <mn>2</mn> <mo>,</mo> <mspace width="negativethinmathspace" /> <mn>3</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1;2,\!3]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db07a3037c8e7f301f996154d79c4970e483bd8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.462ex; height:2.843ex;" alt="{\displaystyle [1;2,\!3]}"></span>。若只把小數點寫成逗號，就會變成<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 [1,2,\!3]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mspace width="negativethinmathspace" /> <mn>3</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1,2,\!3]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ed87bd55e188f4bfbe658c6ccffdfb0fef402b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.462ex; height:2.843ex;" alt="{\displaystyle [1,2,\!3]}"></span>，此時不易判斷究竟是<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 1.2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30b96099eec351cf60cf54ad80159994cec25d61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.176ex;" alt="{\displaystyle 1.2}"></span>與<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 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"></span>之間，還是<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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>與<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 2.3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2.3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2.3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12d28a340d0e52cdf5665a33a969cb7fcb25e014" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.176ex;" alt="{\displaystyle 2.3}"></span>之間的閉區間。 </p> <div class="mw-heading mw-heading2"><h2 id="參考"><span id=".E5.8F.83.E8.80.83"></span>參考</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%8D%80%E9%96%93&action=edit&section=8" title="编辑章节：參考"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20141226211146/http://www.encyclopediaofmath.org/index.php/Interval_and_segment">Interval and segment - Encyclopedia of Mathematics</a>. encyclopediaofmath.org. Springer & The European Mathematical Society. <span class="reference-accessdate"> [<span class="nowrap">2021-05-18</span>]</span>. （<a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Interval_and_segment">原始内容</a>存档于2014-12-26）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%80%E9%96%93&rft.atitle=Interval+and+segment+-+Encyclopedia+of+Mathematics&rft.genre=unknown&rft.jtitle=encyclopediaofmath.org&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FInterval_and_segment&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Vind-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Vind_2-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Vind, Karl. Independence, additivity, uncertainty. Studies in Economic Theory <b>14</b>. Berlin: Springer. 2003. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-3-540-41683-8" title="Special:网络书源/978-3-540-41683-8"><span title="国际标准书号">ISBN</span> 978-3-540-41683-8</a>. <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:1080.91001"><span title="Zentralblatt MATH">Zbl 1080.91001</span></a>. <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-24757-9"><span title="數位物件識別號">doi:10.1007/978-3-540-24757-9</span></a> <span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="连接到英语网页">（英语）</span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%80%E9%96%93&rft.aufirst=Karl&rft.aulast=Vind&rft.btitle=Independence%2C+additivity%2C+uncertainty&rft.date=2003&rft.genre=book&rft.isbn=978-3-540-41683-8&rft.place=Berlin&rft.pub=Springer&rft.series=Studies+in+Economic+Theory&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1080.91001&rft_id=info%3Adoi%2F10.1007%2F978-3-540-24757-9&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Heath-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Heath_3-0">^</a></b></span> <span class="reference-text"><cite class="citation journal">Heath, R. W.; Lutzer, David J.; Zenor, P. L. Monotonically normal spaces. Transactions of the American Mathematical Society. 1973, <b>178</b>: 481–493. <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0002-9947"><span title="国际标准连续出版物号">ISSN 0002-9947</span></a>. <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0372826"><span title="數學評論">MR 0372826</span></a>. <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:0269.54009"><span title="Zentralblatt MATH">Zbl 0269.54009</span></a>. <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1996713"><span title="數位物件識別號">doi:10.2307/1996713</span></a> <span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="连接到英语网页">（英语）</span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%80%E9%96%93&rft.atitle=Monotonically+normal+spaces&rft.au=Lutzer%2C+David+J.&rft.au=Zenor%2C+P.+L.&rft.aufirst=R.+W.&rft.aulast=Heath&rft.date=1973&rft.genre=article&rft.issn=0002-9947&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.pages=481-493&rft.volume=178&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0372826&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0269.54009&rft_id=info%3Adoi%2F10.2307%2F1996713&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20210518074815/https://www.iso.org/standard/3653.html">ISO 31-11:1992</a>. ISO. <span class="reference-accessdate"> [<span class="nowrap">2021-05-18</span>]</span>. （<a rel="nofollow" class="external text" href="https://www.iso.org/standard/3653.html">原始内容</a>存档于2021-05-18） <span style="font-family: sans-serif; cursor: default; color:var(--color-subtle, #54595d); font-size: 0.8em; bottom: 0.1em; font-weight: bold;" title="连接到英语网页">（英语）</span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%80%E9%96%93&rft.atitle=ISO+31-11%3A1992&rft.genre=unknown&rft.jtitle=ISO&rft_id=https%3A%2F%2Fwww.iso.org%2Fstandard%2F3653.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">检索自“<a dir="ltr" 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