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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Đăng nhập</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Trang dành cho người dùng chưa đăng nhập <a href="/wiki/Tr%E1%BB%A3_gi%C3%BAp:Gi%E1%BB%9Bi_thi%E1%BB%87u" aria-label="Tìm hiểu thêm về sửa đổi"><span>tìm hiểu thêm</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:%C4%90%C3%B3ng_g%C3%B3p_c%E1%BB%A7a_t%C3%B4i" title="Danh sách các sửa đổi được thực hiện qua địa chỉ IP này [y]" accesskey="y"><span>Đóng góp</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Th%E1%BA%A3o_lu%E1%BA%ADn_t%C3%B4i" title="Thảo luận với địa chỉ IP này [n]" accesskey="n"><span>Thảo luận cho địa chỉ IP này</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="Trang Web"> <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="Nội dung" 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">Nội dung</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ẩn</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">Đầu</div> </a> </li> <li id="toc-Định_nghĩa_và_ví_dụ" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Định_nghĩa_và_ví_dụ"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Định nghĩa và ví dụ</span> </div> </a> <ul id="toc-Định_nghĩa_và_ví_dụ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Xem_thêm" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Xem_thêm"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Xem thêm</span> </div> </a> <ul id="toc-Xem_thêm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ghi_chú" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ghi_chú"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ghi chú</span> </div> </a> <ul id="toc-Ghi_chú-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tham_khảo" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tham_khảo"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Tham khảo</span> </div> </a> <ul id="toc-Tham_khảo-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="Nội dung" 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="Đóng mở mục lục" > <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">Đóng mở mục lục</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">Không gian đơn liên</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="Xem bài viết trong ngôn ngữ khác. Bài có sẵn trong 27 ngôn ngữ" > <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-27" 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">27 ngôn ngữ</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_terhubung_sederhana" title="Ruang terhubung sederhana – Tiếng Indonesia" lang="id" hreflang="id" data-title="Ruang terhubung sederhana" data-language-autonym="Bahasa Indonesia" data-language-local-name="Tiếng Indonesia" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Conjunt_simplement_connex" title="Conjunt simplement connex – Tiếng Catalan" lang="ca" hreflang="ca" data-title="Conjunt simplement connex" data-language-autonym="Català" data-language-local-name="Tiếng Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%C4%95%D1%80_%C3%A7%D1%8B%D1%85%C4%83%D0%BD%D1%83%D0%BB%D0%BB%C4%83_%D1%83%C3%A7%D0%BB%C4%83%D1%85" title="Пĕр çыхăнуллă уçлăх – Tiếng Chuvash" lang="cv" hreflang="cv" data-title="Пĕр çыхăнуллă уçлăх" data-language-autonym="Чӑвашла" data-language-local-name="Tiếng Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Jednodu%C5%A1e_souvisl%C3%A1_mno%C5%BEina" title="Jednoduše souvislá množina – Tiếng Séc" lang="cs" hreflang="cs" data-title="Jednoduše souvislá množina" data-language-autonym="Čeština" data-language-local-name="Tiếng Séc" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Einfach_zusammenh%C3%A4ngender_Raum" title="Einfach zusammenhängender Raum – Tiếng Đức" lang="de" hreflang="de" data-title="Einfach zusammenhängender Raum" data-language-autonym="Deutsch" data-language-local-name="Tiếng Đức" 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%91%CF%80%CE%BB%CE%AC_%CF%83%CF%85%CE%BD%CE%B5%CE%BA%CF%84%CE%B9%CE%BA%CF%8C%CF%82_%CF%87%CF%8E%CF%81%CE%BF%CF%82" title="Απλά συνεκτικός χώρος – Tiếng Hy Lạp" lang="el" hreflang="el" data-title="Απλά συνεκτικός χώρος" data-language-autonym="Ελληνικά" data-language-local-name="Tiếng Hy Lạp" 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/Simply_connected_space" title="Simply connected space – Tiếng Anh" lang="en" hreflang="en" data-title="Simply connected space" data-language-autonym="English" data-language-local-name="Tiếng Anh" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Conjunto_simplemente_conexo" title="Conjunto simplemente conexo – Tiếng Tây Ban Nha" lang="es" hreflang="es" data-title="Conjunto simplemente conexo" data-language-autonym="Español" data-language-local-name="Tiếng Tây Ban Nha" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Simple_koneksa_spaco" title="Simple koneksa spaco – Tiếng Quốc Tế Ngữ" lang="eo" hreflang="eo" data-title="Simple koneksa spaco" data-language-autonym="Esperanto" data-language-local-name="Tiếng Quốc Tế Ngữ" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D9%87%D9%85%D8%A8%D9%86%D8%AF_%D8%B3%D8%A7%D8%AF%D9%87" title="فضای همبند ساده – Tiếng Ba Tư" lang="fa" hreflang="fa" data-title="فضای همبند ساده" data-language-autonym="فارسی" data-language-local-name="Tiếng Ba Tư" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Connexit%C3%A9_simple" title="Connexité simple – Tiếng Pháp" lang="fr" hreflang="fr" data-title="Connexité simple" data-language-autonym="Français" data-language-local-name="Tiếng Pháp" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A8%EC%9D%BC_%EC%97%B0%EA%B2%B0_%EA%B3%B5%EA%B0%84" title="단일 연결 공간 – Tiếng Hàn" lang="ko" hreflang="ko" data-title="단일 연결 공간" data-language-autonym="한국어" data-language-local-name="Tiếng Hàn" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_semplicemente_connesso" title="Spazio semplicemente connesso – Tiếng Italy" lang="it" hreflang="it" data-title="Spazio semplicemente connesso" data-language-autonym="Italiano" data-language-local-name="Tiếng Italy" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%A4%D7%A9%D7%95%D7%98_%D7%A7%D7%A9%D7%A8" title="מרחב פשוט קשר – Tiếng Do Thái" lang="he" hreflang="he" data-title="מרחב פשוט קשר" data-language-autonym="עברית" data-language-local-name="Tiếng Do Thái" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Enkelvoudig_samenhangende_ruimte" title="Enkelvoudig samenhangende ruimte – Tiếng Hà Lan" lang="nl" hreflang="nl" data-title="Enkelvoudig samenhangende ruimte" data-language-autonym="Nederlands" data-language-local-name="Tiếng Hà Lan" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8D%98%E9%80%A3%E7%B5%90%E7%A9%BA%E9%96%93" title="単連結空間 – Tiếng Nhật" lang="ja" hreflang="ja" data-title="単連結空間" data-language-autonym="日本語" data-language-local-name="Tiếng Nhật" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_jednosp%C3%B3jna" title="Przestrzeń jednospójna – Tiếng Ba Lan" lang="pl" hreflang="pl" data-title="Przestrzeń jednospójna" data-language-autonym="Polski" data-language-local-name="Tiếng Ba Lan" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_simplesmente_conectado" title="Espaço simplesmente conectado – Tiếng Bồ Đào Nha" lang="pt" hreflang="pt" data-title="Espaço simplesmente conectado" data-language-autonym="Português" data-language-local-name="Tiếng Bồ Đào Nha" 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/Spa%C8%9Biu_simplu_conex" title="Spațiu simplu conex – Tiếng Romania" lang="ro" hreflang="ro" data-title="Spațiu simplu conex" data-language-autonym="Română" data-language-local-name="Tiếng Romania" 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%9E%D0%B4%D0%BD%D0%BE%D1%81%D0%B2%D1%8F%D0%B7%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Односвязное пространство – Tiếng Nga" lang="ru" hreflang="ru" data-title="Односвязное пространство" data-language-autonym="Русский" data-language-local-name="Tiếng Nga" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Simply_connected_space" title="Simply connected space – Simple English" lang="en-simple" hreflang="en-simple" data-title="Simply connected space" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Yhdesti_yhten%C3%A4isyys" title="Yhdesti yhtenäisyys – Tiếng Phần Lan" lang="fi" hreflang="fi" data-title="Yhdesti yhtenäisyys" data-language-autonym="Suomi" data-language-local-name="Tiếng Phần Lan" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Enkelt_sammanh%C3%A4ngande_m%C3%A4ngd" title="Enkelt sammanhängande mängd – Tiếng Thụy Điển" lang="sv" hreflang="sv" data-title="Enkelt sammanhängande mängd" data-language-autonym="Svenska" data-language-local-name="Tiếng Thụy Điển" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Basit_ba%C4%9Flant%C4%B1l%C4%B1_uzay" title="Basit bağlantılı uzay – Tiếng Thổ Nhĩ Kỳ" lang="tr" hreflang="tr" data-title="Basit bağlantılı uzay" data-language-autonym="Türkçe" data-language-local-name="Tiếng Thổ Nhĩ Kỳ" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D0%B4%D0%BD%D0%BE%D0%B7%D0%B2%27%D1%8F%D0%B7%D0%BD%D0%B0_%D0%BE%D0%B1%D0%BB%D0%B0%D1%81%D1%82%D1%8C" title="Однозв'язна область – Tiếng Ukraina" lang="uk" hreflang="uk" data-title="Однозв'язна область" data-language-autonym="Українська" data-language-local-name="Tiếng Ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%96%AE%E9%80%A3%E9%80%9A" title="單連通 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="單連通" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%96%AE%E9%80%A3%E9%80%9A" title="單連通 – Tiếng Trung" lang="zh" hreflang="zh" data-title="單連通" data-language-autonym="中文" data-language-local-name="Tiếng Trung" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q912058#sitelinks-wikipedia" title="Sửa liên kết giữa ngôn ngữ" class="wbc-editpage">Sửa liên kết</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Không gian tên"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Kh%C3%B4ng_gian_%C4%91%C6%A1n_li%C3%AAn" title="Xem bài viết [c]" accesskey="c"><span>Bài viết</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Th%E1%BA%A3o_lu%E1%BA%ADn:Kh%C3%B4ng_gian_%C4%91%C6%A1n_li%C3%AAn&action=edit&redlink=1" rel="discussion" class="new" title="Thảo luận về trang này (trang không tồn tại) [t]" accesskey="t"><span>Thảo luận</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Thay đổi biến thể ngôn ngữ" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tiếng Việt</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div 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</nav> <nav class="vector-appearance-landmark" aria-label="Giao diện"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Giao diện</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ẩn</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Bách khoa toàn thư mở Wikipedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="vi" dir="ltr"><p>Trong <a href="/wiki/T%C3%B4_p%C3%B4" title="Tô pô">tô pô</a>, một <a href="/wiki/Kh%C3%B4ng_gian_t%C3%B4_p%C3%B4" title="Không gian tô pô">không gian tôpô</a> được gọi <b>đơn liên</b> nếu nó liên thông đường và <a href="/wiki/Nh%C3%B3m_c%C6%A1_b%E1%BA%A3n" title="Nhóm cơ bản">nhóm cơ bản</a> của nó tại mọi điểm là tầm thường (hay mọi vòng đều <a href="/wiki/%C4%90%E1%BB%93ng_lu%C3%A2n" title="Đồng luân">đồng luân</a> với vòng bất biến). Chẳng hạn, hình tròn (cùng với phần trong của nó) là đơn liên nhưng hình vành khăn là không đơn liên.<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> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Định_nghĩa_và_ví_dụ"><span id=".C4.90.E1.BB.8Bnh_ngh.C4.A9a_v.C3.A0_v.C3.AD_d.E1.BB.A5"></span>Định nghĩa và ví dụ</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kh%C3%B4ng_gian_%C4%91%C6%A1n_li%C3%AAn&veaction=edit&section=1" title="Sửa đổi phần “Định nghĩa và ví dụ”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kh%C3%B4ng_gian_%C4%91%C6%A1n_li%C3%AAn&action=edit&section=1" title="Sửa mã nguồn tại đề mục: Định nghĩa và ví dụ"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/T%E1%BA%ADp_tin:Runge_theorem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Runge_theorem.svg/220px-Runge_theorem.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Runge_theorem.svg/330px-Runge_theorem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Runge_theorem.svg/440px-Runge_theorem.svg.png 2x" data-file-width="800" data-file-height="600" /></a><figcaption>Một không gian <i>không</i> đơn liên: một vòng bao một lỗ thì không thể co về một điểm.</figcaption></figure> <p><a href="/wiki/Kh%C3%B4ng_gian_t%C3%B4_p%C3%B4" title="Không gian tô pô">Không gian tôpô</a> <i>X</i> được gọi <i>đơn liên</i> nếu nó liên thông đường và bất kỳ vòng nào trên <i>X</i> được xác định bởi <i>f</i>: S<sup>1</sup>→<i>X</i> có thể được <a href="/w/index.php?title=Ph%C3%A9p_co_(t%C3%B4_p%C3%B4)&action=edit&redlink=1" class="new" title="Phép co (tô pô) (trang không tồn tại)">co</a> về một điểm. </p><p>Một không gian tô-pô <i>X</i> là đơn liên khi và chỉ khi nó là một không gian <a href="/wiki/T%E1%BA%ADp_h%E1%BB%A3p_li%C3%AAn_th%C3%B4ng#Liên_thông_đường" title="Tập hợp liên thông">liên thông đường</a> và <a href="/wiki/Nh%C3%B3m_c%C6%A1_b%E1%BA%A3n" title="Nhóm cơ bản">nhóm cơ bản</a> của <i>X</i> tại mỗi điểm là tầm thường. </p><p>Tương tự, X là đơn liên khi và chỉ khi với mọi điểm <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d72f66ab332ed430aa9b34ff18c9723c4fea2a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.34ex; height:2.509ex;" alt="{\displaystyle x,y\in X}"></span>, tập hợp các cấu xạ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{\Pi (X)}(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Hom</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{\Pi (X)}(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e56bf408e8e118e592c7586b98aebc2535fb0c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.314ex; height:3.176ex;" alt="{\displaystyle \operatorname {Hom} _{\Pi (X)}(x,y)}"></span> trong <a href="/wiki/Groupoid_c%C6%A1_b%E1%BA%A3n" title="Groupoid cơ bản">groupoid cơ bản</a> của X có đúng một và chỉ một yếu tố.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/T%E1%BA%ADp_tin:P1S2all.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/P1S2all.jpg/400px-P1S2all.jpg" decoding="async" width="400" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/P1S2all.jpg/600px-P1S2all.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/P1S2all.jpg/800px-P1S2all.jpg 2x" data-file-width="1580" data-file-height="309" /></a><figcaption> Một <a href="/wiki/M%E1%BA%B7t_c%E1%BA%A7u" title="Mặt cầu">hình cầu</a> là một không gian đơn liên: mọi vòng đều có thể co về một điểm.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/T%E1%BA%ADp_tin:Torus_cycles.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Torus_cycles.png/150px-Torus_cycles.png" decoding="async" width="150" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Torus_cycles.png/225px-Torus_cycles.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/54/Torus_cycles.png 2x" data-file-width="229" data-file-height="311" /></a><figcaption> Một hình xuyến không phải là một bề mặt đơn liên. Cả hai vòng màu tím đều không thể co về một điểm.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Xem_thêm"><span id="Xem_th.C3.AAm"></span>Xem thêm</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kh%C3%B4ng_gian_%C4%91%C6%A1n_li%C3%AAn&veaction=edit&section=2" title="Sửa đổi phần “Xem thêm”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kh%C3%B4ng_gian_%C4%91%C6%A1n_li%C3%AAn&action=edit&section=2" title="Sửa mã nguồn tại đề mục: Xem thêm"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Nh%C3%B3m_c%C6%A1_b%E1%BA%A3n" title="Nhóm cơ bản">Nhóm cơ bản</a></li> <li><a href="/wiki/Groupoid_c%C6%A1_b%E1%BA%A3n" title="Groupoid cơ bản">Groupoid cơ bản</a></li> <li><a href="/w/index.php?title=Ph%C3%A9p_co_(t%C3%B4_p%C3%B4)&action=edit&redlink=1" class="new" title="Phép co (tô pô) (trang không tồn tại)">Biến dạng co</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Ghi_chú"><span id="Ghi_ch.C3.BA"></span>Ghi chú</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kh%C3%B4ng_gian_%C4%91%C6%A1n_li%C3%AAn&veaction=edit&section=3" title="Sửa đổi phần “Ghi chú”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kh%C3%B4ng_gian_%C4%91%C6%A1n_li%C3%AAn&action=edit&section=3" title="Sửa mã nguồn tại đề mục: Ghi chú"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r71728118">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text">Đoàn Quỳnh (2000), tr. 291</span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text">Ronald, Brown (2006)</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Tham_khảo"><span id="Tham_kh.E1.BA.A3o"></span>Tham khảo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kh%C3%B4ng_gian_%C4%91%C6%A1n_li%C3%AAn&veaction=edit&section=4" title="Sửa đổi phần “Tham khảo”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kh%C3%B4ng_gian_%C4%91%C6%A1n_li%C3%AAn&action=edit&section=4" title="Sửa mã nguồn tại đề mục: Tham khảo"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r67233549">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"“""”""‘""’"}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFSpanier1994" class="citation book cs1">Spanier, Edwin (tháng 12 năm 1994). <i>Algebraic Topology</i>. Springer. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/0-387-94426-5" title="Đặc biệt:Nguồn sách/0-387-94426-5"><bdi>0-387-94426-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Topology&rft.pub=Springer&rft.date=1994-12&rft.isbn=0-387-94426-5&rft.aulast=Spanier&rft.aufirst=Edwin&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AKh%C3%B4ng+gian+%C4%91%C6%A1n+li%C3%AAn" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFConway1986" class="citation book cs1">Conway, John (1986). <i>Functions of One Complex Variable I</i>. Springer. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/0-387-90328-3" title="Đặc biệt:Nguồn sách/0-387-90328-3"><bdi>0-387-90328-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functions+of+One+Complex+Variable+I&rft.pub=Springer&rft.date=1986&rft.isbn=0-387-90328-3&rft.aulast=Conway&rft.aufirst=John&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AKh%C3%B4ng+gian+%C4%91%C6%A1n+li%C3%AAn" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFBourbaki2005" class="citation book cs1">Bourbaki, Nicolas (2005). <i>Lie Groups and Lie Algebras</i>. Springer. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/3-540-43405-4" title="Đặc biệt:Nguồn sách/3-540-43405-4"><bdi>3-540-43405-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+Groups+and+Lie+Algebras&rft.pub=Springer&rft.date=2005&rft.isbn=3-540-43405-4&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AKh%C3%B4ng+gian+%C4%91%C6%A1n+li%C3%AAn" class="Z3988"></span></li> <li>Đoàn Quỳnh, 2000, <i>Hình học vi phân</i></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFGamelin2001" class="citation book cs1">Gamelin, Theodore (tháng 1 năm 2001). <i>Complex Analysis</i>. Springer. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/0-387-95069-9" title="Đặc biệt:Nguồn sách/0-387-95069-9"><bdi>0-387-95069-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+Analysis&rft.pub=Springer&rft.date=2001-01&rft.isbn=0-387-95069-9&rft.aulast=Gamelin&rft.aufirst=Theodore&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AKh%C3%B4ng+gian+%C4%91%C6%A1n+li%C3%AAn" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFJoshi1983" class="citation book cs1">Joshi, Kapli (tháng 8 năm 1983). <i>Introduction to General Topology</i>. New Age Publishers. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/0-85226-444-5" title="Đặc biệt:Nguồn sách/0-85226-444-5"><bdi>0-85226-444-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+General+Topology&rft.pub=New+Age+Publishers&rft.date=1983-08&rft.isbn=0-85226-444-5&rft.aulast=Joshi&rft.aufirst=Kapli&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AKh%C3%B4ng+gian+%C4%91%C6%A1n+li%C3%AAn" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r67233549"><cite id="CITEREFRonald2006" class="citation book cs1">Ronald, Brown (tháng 6 năm 2006). <i>Topology and Groupoids</i>. Academic Search Complete. North Charleston: CreateSpace. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/1419627228" title="Đặc biệt:Nguồn sách/1419627228"><bdi>1419627228</bdi></a>. <a href="/wiki/OCLC_(%C4%91%E1%BB%8Bnh_danh)" class="mw-redirect" title="OCLC (định danh)">OCLC</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/oclc/712629429">712629429</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology+and+Groupoids.&rft.place=North+Charleston&rft.pub=CreateSpace&rft.date=2006-06&rft_id=info%3Aoclcnum%2F712629429&rft.isbn=1419627228&rft.aulast=Ronald&rft.aufirst=Brown&rfr_id=info%3Asid%2Fvi.wikipedia.org%3AKh%C3%B4ng+gian+%C4%91%C6%A1n+li%C3%AAn" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Lấy từ “<a dir="ltr" href="https://vi.wikipedia.org/w/index.php?title=Không_gian_đơn_liên&oldid=63188622">https://vi.wikipedia.org/w/index.php?title=Không_gian_đơn_liên&oldid=63188622</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Th%E1%BB%83_lo%E1%BA%A1i" title="Đặc biệt:Thể loại">Thể loại</a>: <ul><li><a href="/wiki/Th%E1%BB%83_lo%E1%BA%A1i:T%C3%B4_p%C3%B4_%C4%91%E1%BA%A1i_s%E1%BB%91" title="Thể loại:Tô pô đại số">Tô pô đại số</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Trang này được sửa đổi lần cuối vào ngày 4 tháng 8 năm 2020, 15:35.</li> <li id="footer-info-copyright">Văn bản được phát hành theo <a href="/wiki/Wikipedia:Nguy%C3%AAn_v%C4%83n_Gi%E1%BA%A5y_ph%C3%A9p_Creative_Commons_Ghi_c%C3%B4ng%E2%80%93Chia_s%E1%BA%BB_t%C6%B0%C6%A1ng_t%E1%BB%B1_phi%C3%AAn_b%E1%BA%A3n_4.0_Qu%E1%BB%91c_t%E1%BA%BF" title="Wikipedia:Nguyên văn Giấy phép Creative Commons Ghi công–Chia sẻ tương tự phiên bản 4.0 Quốc tế">Giấy phép Creative Commons Ghi công–Chia sẻ tương tự</a>; có thể áp dụng điều khoản bổ sung. 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