CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-disabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-not-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>View source for Directed set - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-disabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-not-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":true,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat": "dmy","wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"7fb68c6d-1f21-4d34-b83b-323eec0603d9","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Directed_set","wgTitle":"Directed set","wgCurRevisionId":1258015664,"wgRevisionId":0,"wgArticleId":9165,"wgIsArticle":false,"wgIsRedirect":false,"wgAction":"edit","wgUserName":null,"wgUserGroups":["*"],"wgCategories":[],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Directed_set","wgRelevantArticleId":9165,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{ "pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","skins.vector.search.codex.styles":"ready" ,"skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.charinsert.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["mediawiki.action.edit.collapsibleFooter","site","mediawiki.page.ready","jquery.makeCollapsible","skins.vector.js","ext.centralNotice.geoIP","ext.charinsert","ext.gadget.ReferenceTooltips","ext.gadget.charinsert","ext.gadget.extra-toolbar-buttons","ext.gadget.refToolbar","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession", "wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.charinsert.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="noindex,nofollow,max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="View source for Directed set - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Directed_set"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Directed_set&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Directed_set"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Directed_set rootpage-Directed_set skin-vector-2022 action-edit"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Directed+set&returntoquery=action%3Dedit%26undo%3D1193248815%26undoafter%3D1176830014" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Directed+set&returntoquery=action%3Dedit%26undo%3D1193248815%26undoafter%3D1176830014" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Directed+set&returntoquery=action%3Dedit%26undo%3D1193248815%26undoafter%3D1176830014" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Directed+set&returntoquery=action%3Dedit%26undo%3D1193248815%26undoafter%3D1176830014" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</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"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</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/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</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="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <h1 id="firstHeading" class="firstHeading mw-first-heading">View source for Directed set</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="This article exist only in this language. Add the article for other languages" > <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-0" 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">Add languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> <div class="after-portlet after-portlet-lang"><span class="uls-after-portlet-link"></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <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/Directed_set" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Directed_set" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</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="Change language variant" > <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">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="vector-tab-noicon mw-list-item"><a href="/wiki/Directed_set"><span>Read</span></a></li><li id="ca-edit" class="selected vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Directed_set&action=edit" title="Edit this page"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Directed_set&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-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">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="vector-more-collapsible-item mw-list-item"><a href="/wiki/Directed_set"><span>Read</span></a></li><li id="ca-more-edit" class="selected vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Directed_set&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Directed_set&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Directed_set" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Directed_set" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Directed_set&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fw%2Findex.php%3Ftitle%3DDirected_set%26action%3Dedit%26undo%3D1193248815%26undoafter%3D1176830014"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fw%2Findex.php%3Ftitle%3DDirected_set%26action%3Dedit%26undo%3D1193248815%26undoafter%3D1176830014"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1513048" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <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">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</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> <div id="contentSub"><div id="mw-content-subtitle">← <a href="/wiki/Directed_set" title="Directed set">Directed set</a></div></div> <div id="mw-content-text" class="mw-body-content"><div class="error mw-undo-failure"><div class="mw-parser-output"> <style data-mw-deduplicate="TemplateStyles:r1238441935">.mw-parser-output .fmbox{clear:both;margin:0.2em 0;width:100%;border:1px solid #a2a9b1;background-color:var(--background-color-interactive-subtle,#f8f9fa);box-sizing:border-box;color:var(--color-base,#202122)}.mw-parser-output .fmbox-warning{border:1px solid #bb7070;background-color:#ffdbdb}.mw-parser-output .fmbox-editnotice{background-color:transparent}.mw-parser-output .fmbox .mbox-text{border:none;padding:0.25em 0.9em;width:100%}.mw-parser-output .fmbox .mbox-image{border:none;padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .fmbox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .fmbox .mbox-invalid-type{text-align:center}@media screen{html.skin-theme-clientpref-night .mw-parser-output .fmbox-warning{background-color:#300}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .fmbox-warning{background-color:#300}}</style><table id="mw-undo-failure" class="plainlinks fmbox fmbox-warning" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/Help:Undo" title="Help:Undo"><img src="//upload.wikimedia.org/wikipedia/en/thumb/2/2c/Nuvola_actions_undo.png/40px-Nuvola_actions_undo.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/2/2c/Nuvola_actions_undo.png 1.5x" data-file-width="48" data-file-height="48" /></a></span></td><td class="mbox-text">The edit could not be <a href="/wiki/Help:Undo" class="mw-redirect" title="Help:Undo">undone</a> due to conflicting intermediate edits; if you wish to undo the change, it must be <a href="/wiki/Help:Reverting" title="Help:Reverting">done manually</a>.</td></tr></tbody></table> </div></div><p>You do not have permission to edit this page, for the following reasons: </p> <ul class="permissions-errors"><li class="mw-permissionerror-blockedtext"> <div id="mw-blocked-text" style="border: 1px solid #AAA; background-color: var(--background-color-warning-subtle, ivory); color: inherit; padding: 1.5em; width: 100%; box-sizing: border-box;"> <div style="text-align: center;"><span style="font-size: 26px;"><span typeof="mw:File"><a href="/wiki/File:Stop_hand_nuvola.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/50px-Stop_hand_nuvola.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/75px-Stop_hand_nuvola.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/100px-Stop_hand_nuvola.svg.png 2x" data-file-width="240" data-file-height="240" /></a></span><b> This IP address has been <a href="/wiki/Wikipedia:Blocking_policy" title="Wikipedia:Blocking policy">blocked</a> from <i>editing</i> Wikipedia.</b></span><br /><span style="font-size: 18px;">This does not affect your ability to <i>read</i> Wikipedia pages.</span></div><div class="paragraphbreak" style="margin-top:0.5em"></div><b>Most people who see this message have done nothing wrong.</b> Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.<div class="paragraphbreak" style="margin-top:0.5em"></div> <p>The IP address or range 8.222.128.0/17 has been <a href="/wiki/Wikipedia:Blocking_policy" title="Wikipedia:Blocking policy">blocked</a> by <a href="/wiki/User:L235" title="User:L235">‪L235‬</a> for the following reason(s): </p> <div style="padding:10px; background:var(--background-color-base, white); color:inherit; border:1px #666 solid;"> <div class="user-block colocation-webhost" style="margin-bottom: 0.5em; background-color: #ffefd5; border: 1px solid #AAA; padding: 0.7em;"> <figure class="mw-halign-left" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Server-multiple.svg/40px-Server-multiple.svg.png" decoding="async" width="40" height="57" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Server-multiple.svg/60px-Server-multiple.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Server-multiple.svg/80px-Server-multiple.svg.png 2x" data-file-width="744" data-file-height="1052" /></span><figcaption></figcaption></figure><b>The <a href="/wiki/IP_address" title="IP address">IP address</a> that you are currently using has been blocked because it is believed to be a <a href="/wiki/Web_hosting_service" title="Web hosting service">web host provider</a> or <a href="/wiki/Colocation_centre" title="Colocation centre">colocation provider</a>.</b> To prevent abuse, <a href="/wiki/Wikipedia:Open_proxies" title="Wikipedia:Open proxies">web hosts and colocation providers may be blocked</a> from editing Wikipedia. <div style="border-top: 1px solid #AAA; clear: both">You will not be able to edit Wikipedia using a web host or colocation provider because it hides your IP address, much like a <a href="/wiki/Wikipedia:Open_proxies" title="Wikipedia:Open proxies">proxy</a> or <a href="/wiki/Virtual_private_network" title="Virtual private network">VPN</a>. <p><b>We recommend that you attempt to use another connection to edit.</b> For example, if you use a proxy or VPN to connect to the internet, turn it off when editing Wikipedia. If you edit using a mobile connection, try using a Wi-Fi connection, and vice versa. If you are using a corporate internet connection, switch to a different Wi-Fi network. If you have a Wikipedia account, please log in. </p><p>If you do not have any other way to edit Wikipedia, you will need to <a href="/wiki/Wikipedia:IP_block_exemption#Requesting_and_granting_exemption" title="Wikipedia:IP block exemption">request an IP block exemption</a>. </p> <style data-mw-deduplicate="TemplateStyles:r1214851843">.mw-parser-output .hidden-begin{box-sizing:border-box;width:100%;padding:5px;border:none;font-size:95%}.mw-parser-output .hidden-title{font-weight:bold;line-height:1.6;text-align:left}.mw-parser-output .hidden-content{text-align:left}@media all and (max-width:500px){.mw-parser-output .hidden-begin{width:auto!important;clear:none!important;float:none!important}}</style><div class="hidden-begin mw-collapsible mw-collapsed" style=""><div class="hidden-title skin-nightmode-reset-color" style="text-align:center;">How to appeal if you are confident that your connection does not use a colocation provider's IP address:</div><div class="hidden-content mw-collapsible-content" style=""> If you are confident that you are not using a web host, you may <a href="/wiki/Wikipedia:Appealing_a_block" title="Wikipedia:Appealing a block">appeal this block</a> by adding the following text on your <a href="/wiki/Help:Talk_pages" title="Help:Talk pages">talk page</a>: <code>{{<a href="/wiki/Template:Unblock" title="Template:Unblock">unblock</a>|reason=Caught by a colocation web host block but this host or IP is not a web host. My IP address is _______. <i>Place any further information here.</i> ~~~~}}</code>. <b>You must fill in the blank with your IP address for this block to be investigated.</b> Your IP address can be determined <span class="plainlinks"><b><a class="external text" href="https://en.wikipedia.org/wiki/Wikipedia:Get_my_IP_address?withJS=MediaWiki:Get-my-ip.js">here</a></b></span>. Alternatively, if you wish to keep your IP address private you can use the <a href="/wiki/Wikipedia:Unblock_Ticket_Request_System" title="Wikipedia:Unblock Ticket Request System">unblock ticket request system</a>. There are several reasons you might be editing using the IP address of a web host or colocation provider (such as if you are using VPN software or a business network); please use this method of appeal only if you think your IP address is in fact not a web host or colocation provider.</div></div> <p><span class="sysop-show" style="font-size: 85%;"><span style="border:#707070 solid 1px;background-color:#ffe0e0;padding:2px"><b>Administrators:</b></span> The <a href="/wiki/Wikipedia:IP_block_exemption" title="Wikipedia:IP block exemption">IP block exemption</a> user right should only be applied to allow users to edit using web host in exceptional circumstances, and requests should usually be directed to the functionaries team via email. If you intend to give the IPBE user right, a <a href="/wiki/Wikipedia:CheckUser" title="Wikipedia:CheckUser">CheckUser</a> needs to take a look at the account. This can be requested most easily at <a href="/wiki/Wikipedia:SPI#Quick_CheckUser_requests" class="mw-redirect" title="Wikipedia:SPI">SPI Quick Checkuser Requests</a>. <b>Unblocking</b> an IP or IP range with this template <b>is highly discouraged</b> without at least contacting the blocking administrator.</span> </p> </div></div> </div> <p>This block will expire on 18:23, 24 August 2026. Your current IP address is 8.222.208.146. </p> <div class="paragraphbreak" style="margin-top:0.5em"></div><div style="font-size: 16px;"> <p>Even when blocked, you will <i>usually</i> still be able to edit your <a href="/wiki/Special:MyTalk" title="Special:MyTalk">user talk page</a>, as well as <a href="/wiki/Wikipedia:Emailing_users" title="Wikipedia:Emailing users">email</a> administrators and other editors. </p> </div> <div class="paragraphbreak" style="margin-top:0.5em"></div><div style="font-size: 16px;"> <p>For information on how to proceed, please read the <b><a href="/wiki/Wikipedia:Appealing_a_block#Common_questions" title="Wikipedia:Appealing a block">FAQ for blocked users</a></b> and the <a href="/wiki/Wikipedia:Appealing_a_block" title="Wikipedia:Appealing a block">guideline on block appeals</a>. The <a href="/wiki/Wikipedia:Guide_to_appealing_blocks" title="Wikipedia:Guide to appealing blocks">guide to appealing blocks</a> may also be helpful. </p> </div> <p>Other useful links: <a href="/wiki/Wikipedia:Blocking_policy" title="Wikipedia:Blocking policy">Blocking policy</a> · <a href="/wiki/Help:I_have_been_blocked" title="Help:I have been blocked">Help:I have been blocked</a> </p> </div></li><li class="mw-permissionerror-globalblocking-blockedtext-range"> <div id="mw-blocked-text" style="border: 1px solid #AAA; background-color: var(--background-color-warning-subtle, ivory); color: inherit; padding: 1.5em; width: 100%; box-sizing: border-box;"> <div style="text-align: center;"><span style="font-size: 26px;"><span typeof="mw:File"><a href="/wiki/File:Stop_hand_nuvola.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/50px-Stop_hand_nuvola.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/75px-Stop_hand_nuvola.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f1/Stop_hand_nuvola.svg/100px-Stop_hand_nuvola.svg.png 2x" data-file-width="240" data-file-height="240" /></a></span><b> This IP address range has been <a href="https://meta.wikimedia.org/wiki/Global_blocks" class="extiw" title="m:Global blocks">globally blocked</a>.</b></span><br /><span style="font-size: 18px;">This does not affect your ability to <i>read</i> Wikipedia pages.</span></div><div class="paragraphbreak" style="margin-top:0.5em"></div><b>Most people who see this message have done nothing wrong.</b> Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.<div class="paragraphbreak" style="margin-top:0.5em"></div><div class="paragraphbreak" style="margin-top:0.5em"></div> <p>This block affects editing on all Wikimedia wikis. </p><p>The IP address or range 8.222.128.0/17 has been globally <a href="/wiki/Wikipedia:Blocking_policy" title="Wikipedia:Blocking policy">blocked</a> by <a href="/wiki/User:Jon_Kolbert" title="User:Jon Kolbert">‪Jon Kolbert‬</a> for the following reason(s): </p> <div style="padding:10px; background:var(--background-color-base, white); color:inherit; border:1px #666 solid;"> <p><a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/NOP" class="extiw" title="m:Special:MyLanguage/NOP">Open proxy/Webhost</a>: See the <a href="https://meta.wikimedia.org/wiki/WM:OP/H" class="extiw" title="m:WM:OP/H">help page</a> if you are affected </p> </div> <p>This block will expire on 15:12, 27 August 2028. Your current IP address is 8.222.208.146. </p> <div class="paragraphbreak" style="margin-top:0.5em"></div><div style="font-size: 16px;"> <p>Even while globally blocked, you will <i>usually</i> still be able to edit pages on <a href="https://meta.wikimedia.org/wiki/" class="extiw" title="m:">Meta-Wiki</a>. </p> </div> <div class="paragraphbreak" style="margin-top:0.5em"></div><div style="font-size: 16px;"> <p>If you believe you were blocked by mistake, you can find additional information and instructions in the <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/No_open_proxies" class="extiw" title="m:Special:MyLanguage/No open proxies">No open proxies</a> global policy. Otherwise, to discuss the block please <a href="https://meta.wikimedia.org/wiki/Steward_requests/Global" class="extiw" title="m:Steward requests/Global">post a request for review on Meta-Wiki</a>. You could also send an email to the <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/Stewards" class="extiw" title="m:Special:MyLanguage/Stewards">stewards</a> <a href="https://meta.wikimedia.org/wiki/Special:MyLanguage/VRT" class="extiw" title="m:Special:MyLanguage/VRT">VRT</a> queue at <kbd>stewards@wikimedia.org</kbd> including all above details. </p> </div> <p>Other useful links: <a href="https://meta.wikimedia.org/wiki/Global_blocks" class="extiw" title="m:Global blocks">Global blocks</a> · <a href="/wiki/Help:I_have_been_blocked" title="Help:I have been blocked">Help:I have been blocked</a> </p> </div></li></ul><hr /> <div id="viewsourcetext">You can view and copy the source of this page:</div><textarea readonly="" accesskey="," id="wpTextbox1" cols="80" rows="25" style="" class="mw-editfont-monospace" lang="en" dir="ltr" name="wpTextbox1">{{Short description|Mathematical ordering with upper bounds}} In [[mathematics]], a '''directed set''' (or a '''directed preorder''' or a '''filtered set''') is a nonempty [[Set (mathematics)|set]] <math>A</math> together with a [[Reflexive relation|reflexive]] and [[Transitive relation|transitive]] [[binary relation]] <math>\,\leq\,</math> (that is, a [[preorder]]), with the additional property that every pair of elements has an [[upper bound]].{{sfn|Kelley|1975|pp=65}} In other words, for any <math>a</math> and <math>b</math> in <math>A</math> there must exist <math>c</math> in <math>A</math> with <math>a \leq c</math> and <math>b \leq c.</math> A directed set's preorder is called a '''direction'''. The notion defined above is sometimes called an '''{{visible anchor|upward directed set}}'''. A '''{{visible anchor|downward directed set}}''' is defined analogously,<ref>{{cite book|author=Robert S. Borden|title=A Course in Advanced Calculus|year=1988|publisher=Courier Corporation|isbn=978-0-486-15038-3|page=20}}</ref> meaning that every pair of elements is bounded below.<ref name="Brown-Pearcy">{{cite book|author1=Arlen Brown|author2=Carl Pearcy|title=An Introduction to Analysis|url=https://archive.org/details/introductiontoan0000brow|url-access=registration|year=1995|publisher=Springer|isbn=978-1-4612-0787-0|page=[https://archive.org/details/introductiontoan0000brow/page/13 13]}}</ref> Some authors (and this article) assume that a directed set is directed upward, unless otherwise stated. Other authors call a set directed if and only if it is directed both upward and downward.<ref name="CarlHeikkilä2010">{{cite book|author1=Siegfried Carl|author2=Seppo Heikkilä|title=Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory|year=2010|publisher=Springer|isbn=978-1-4419-7585-0|pages=77}}</ref> Directed sets are a generalization of nonempty [[totally ordered set]]s. That is, all totally ordered sets are directed sets (contrast [[Partially ordered sets|{{em|partially}} ordered sets]], which need not be directed). [[Join-semilattice]]s (which are partially ordered sets) are directed sets as well, but not conversely. Likewise, [[Lattice (order)|lattice]]s are directed sets both upward and downward. In [[topology]], directed sets are used to define [[Net (topology)|nets]], which generalize [[sequence]]s and unite the various notions of [[Limit (mathematics)|limit]] used in [[Mathematical analysis|analysis]]. Directed sets also give rise to [[direct limit]]s in [[abstract algebra]] and (more generally) [[category theory]]. ==Equivalent definition== In addition to the definition above, there is an equivalent definition. A '''directed set''' is a set <math>A</math> with a [[preorder]] such that every finite subset of <math>A</math> has an upper bound. In this definition, the existence of an upper bound of the [[Empty set|empty subset]] implies that <math>A</math> is nonempty. ==Examples== The set of [[natural number]]s <math>\N</math> with the ordinary order <math>\,\leq\,</math> is one of the most important examples of a directed set. Every [[Total order|totally ordered set]] is a directed set, including <math>(\N, \leq),</math> <math>(\N, \geq),</math> <math>(\Reals, \leq),</math> and <math>(\Reals, \geq).</math> A (trivial) example of a partially ordered set that is '''{{em|not}}''' directed is the set <math>\{a, b\},</math> in which the only order relations are <math>a \leq a</math> and <math>b \leq b.</math> A less trivial example is like the following example of the "reals directed towards <math>x_0</math>" but in which the ordering rule only applies to pairs of elements on the same side of <math>x_0</math> (that is, if one takes an element <math>a</math> to the left of <math>x_0,</math> and <math>b</math> to its right, then <math>a</math> and <math>b</math> are not comparable, and the subset <math>\{ a, b \}</math> has no upper bound). ===Product of directed sets=== Let <math>\mathbb{D}_1</math> and <math>\mathbb{D}_2</math> be directed sets. Then the [[Cartesian product]] set <math>\mathbb{D}_1 \times \mathbb{D}_2</math> can be made into a directed set by defining <math>\left(n_1, n_2\right) \leq \left(m_1, m_2\right)</math> if and only if <math>n_1 \leq m_1</math> and <math>n_2 \leq m_2.</math> In analogy to the [[product order]] this is the product direction on the Cartesian product. For example, the set <math>\N \times \N</math> of pairs of natural numbers can be made into a directed set by defining <math>\left(n_0, n_1\right) \leq \left(m_0, m_1\right)</math> if and only if <math>n_0 \leq m_0</math> and <math>n_1 \leq m_1.</math> ===Directed towards a point=== If <math>x_0</math> is a [[real number]] then the set <math>I := \R \backslash \lbrace x_0 \rbrace</math> can be turned into a directed set by defining <math>a \leq_I b</math> if <math>\left|a - x_0\right| \geq \left|b - x_0\right|</math> (so "greater" elements are closer to <math>x_0</math>). We then say that the reals have been '''directed towards <math>x_0.</math>''' This is an example of a directed set that is {{em|neither}} [[Partial order|partially ordered]] nor [[Total order|totally ordered]]. This is because [[Antisymmetric relation|antisymmetry]] breaks down for every pair <math>a</math> and <math>b</math> equidistant from <math>x_0,</math> where <math>a</math> and <math>b</math> are on opposite sides of <math>x_0.</math> Explicitly, this happens when <math>\{a, b\} = \left\{x_0 - r, x_0 + r\right\}</math> for some real <math>r \neq 0,</math> in which case <math>a \leq_I b</math> and <math>b \leq_I a</math> even though <math>a \neq b.</math> Had this preorder been defined on <math>\R</math> instead of <math>\R \backslash \lbrace x_0 \rbrace</math> then it would still form a directed set but it would now have a (unique) [[greatest element]], specifically <math>x_0</math>; however, it still wouldn't be partially ordered. This example can be generalized to a [[metric space]] <math>(X, d)</math> by defining on <math>X</math> or <math>X \setminus \left\{x_0\right\}</math> the preorder <math>a \leq b</math> if and only if <math>d\left(a, x_0\right) \geq d\left(b, x_0\right).</math> ===Maximal and greatest elements=== An element <math>m</math> of a preordered set <math>(I, \leq)</math> is a ''[[Maximal and minimal elements|maximal element]]'' if for every <math>j \in I,</math> <math>m \leq j</math> implies <math>j \leq m.</math><ref>This implies <math>j = m</math> if <math>(I, \leq)</math> is a [[partially ordered set]].</ref> It is a ''[[Greatest element and least element|greatest element]]'' if for every <math>j \in I,</math> <math>j \leq m.</math> Any preordered set with a greatest element is a directed set with the same preorder. For instance, in a [[poset]] <math>P,</math> every [[Upper set#Upper closure and lower closure|lower closure]] of an element; that is, every subset of the form <math>\{a \in P : a \leq x\}</math> where <math>x</math> is a fixed element from <math>P,</math> is directed. Every maximal element of a directed preordered set is a greatest element. Indeed, a directed preordered set is characterized by equality of the (possibly empty) sets of maximal and of greatest elements. ===Subset inclusion=== The [[subset inclusion]] relation <math>\,\subseteq,\,</math> along with its [[Duality (order theory)|dual]] <math>\,\supseteq,\,</math> define [[partial order]]s on any given [[family of sets]]. A non-empty [[family of sets]] is a directed set with respect to the partial order <math>\,\supseteq\,</math> (respectively, <math>\,\subseteq\,</math>) if and only if the intersection (respectively, union) of any two of its members contains as a subset (respectively, is contained as a subset of) some third member. In symbols, a family <math>I</math> of sets is directed with respect to <math>\,\supseteq\,</math> (respectively, <math>\,\subseteq\,</math>) if and only if :for all <math>A, B \in I,</math> there exists some <math>C \in I</math> such that <math>A \supseteq C</math> and <math>B \supseteq C</math> (respectively, <math>A \subseteq C</math> and <math>B \subseteq C</math>) or equivalently, :for all <math>A, B \in I,</math> there exists some <math>C \in I</math> such that <math>A \cap B \supseteq C</math> (respectively, <math>A \cup B \subseteq C</math>). Many important examples of directed sets can be defined using these partial orders. For example, by definition, a [[Filter (set theory)|{{em|prefilter}}]] or {{em|filter base}} is a non-empty [[family of sets]] that is a directed set with respect to the [[partial order]] <math>\,\supseteq\,</math> and that also does not contain the empty set (this condition prevents triviality because otherwise, the empty set would then be a [[Greatest element and least element|greatest element]] with respect to <math>\,\supseteq\,</math>). Every [[Pi-system|{{pi}}-system]], which is a non-empty [[family of sets]] that is closed under the intersection of any two of its members, is a directed set with respect to <math>\,\supseteq\,.</math> Every [[Dynkin system|λ-system]] is a directed set with respect to <math>\,\subseteq\,.</math> Every [[Filter (set theory)|filter]], [[Topology (structure)|topology]], and [[σ-algebra]] is a directed set with respect to both <math>\,\supseteq\,</math> and <math>\,\subseteq\,.</math> ====Tails of nets==== By definition, a {{em|[[Net (mathematics)|net]]}} is a function from a directed set and a [[Sequence (mathematics)|sequence]] is a function from the natural numbers <math>\N.</math> Every sequence canonically becomes a net by endowing <math>\N</math> with <math>\,\leq.\,</math> If <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> is any [[Net (mathematics)|net]] from a directed set <math>(I, \leq)</math> then for any index <math>i \in I,</math> the set <math>x_{\geq i} := \left\{x_j : j \geq i \text{ with } j \in I\right\}</math> is called the tail of <math>(I, \leq)</math> starting at <math>i.</math> The family <math>\operatorname{Tails}\left(x_{\bull}\right) := \left\{x_{\geq i} : i \in I\right\}</math> of all tails is a directed set with respect to <math>\,\supseteq;\,</math> in fact, it is even a prefilter. ====Neighborhoods==== If <math>T</math> is a [[topological space]] and <math>x_0</math> is a point in <math>T,</math> the set of all [[Topological neighbourhood|neighbourhoods]] of <math>x_0</math> can be turned into a directed set by writing <math>U \leq V</math> if and only if <math>U</math> contains <math>V.</math> For every <math>U,</math> <math>V,</math> and <math>W</math>{{hairsp}}: * <math>U \leq U</math> since <math>U</math> contains itself. * if <math>U \leq V</math> and <math>V \leq W,</math> then <math>U \supseteq V</math> and <math>V \supseteq W,</math> which implies <math>U \supseteq W.</math> Thus <math>U \leq W.</math> * because <math>x_0 \in U \cap V,</math> and since both <math>U \supseteq U \cap V</math> and <math>V \supseteq U \cap V,</math> we have <math>U \leq U \cap V</math> and <math>V \leq U \cap V.</math> ====Finite subsets==== The set <math>\operatorname{Finite}(I)</math> of all finite subsets of a set <math>I</math> is directed with respect to <math>\,\subseteq\,</math> since given any two <math>A, B \in \operatorname{Finite}(I),</math> their union <math>A \cup B \in \operatorname{Finite}(I)</math> is an upper bound of <math>A</math> and <math>B</math> in <math>\operatorname{Finite}(I).</math> This particular directed set is used to define the sum <math>{\textstyle\sum\limits_{i \in I}} r_i</math> of a [[Generalized series (mathematics)|generalized series]] of an <math>I</math>-indexed collection of numbers <math>\left(r_i\right)_{i \in I}</math> (or more generally, the sum of [[Series (mathematics)#Abelian topological groups|elements in an]] [[abelian topological group]], such as [[Series (mathematics)#Series in topological vector spaces|vectors]] in a [[topological vector space]]) as the [[Limit of a net|limit of the net]] of [[partial sum]]s <math>F \in \operatorname{Finite}(I) \mapsto {\textstyle\sum\limits_{i \in F}} r_i;</math> that is: <math display=block>\sum_{i \in I} r_i ~:=~ \lim_{F \in \operatorname{Finite}(I)} \ \sum_{i \in F} r_i ~=~ \lim \left\{\sum_{i \in F} r_i \,: F \subseteq I, F \text{ finite }\right\}.</math> ===Logic=== {{See also|Preorder#Preorders and partial orders on partitions}} Let <math>S</math> be a [[Theory (mathematical logic)|formal theory]], which is a set of [[Sentence (mathematical logic)|sentences]] with certain properties (details of which can be found in [[Theory (mathematical logic)|the article on the subject]]). For instance, <math>S</math> could be a [[first-order theory]] (like [[Zermelo–Fraenkel set theory]]) or a simpler [[Propositional calculus|zeroth-order theory]]. The preordered set <math>(S, \Leftarrow)</math> is a directed set because if <math>A, B \in S</math> and if <math>C := A \wedge B</math> denotes the sentence formed by [[logical conjunction]] <math>\,\wedge,\,</math> then <math>A \Leftarrow C</math> and <math>B \Leftarrow C</math> where <math>C \in S.</math> If <math>S / \sim</math> is the [[Lindenbaum–Tarski algebra]] associated with <math>S</math> then <math>\left(S / \sim, \Leftarrow\right)</math> is a partially ordered set that is also a directed set. ==Contrast with semilattices== [[File:Directed_set,_but_no_join_semi-lattice.png|thumb|x100px|Example of a directed set which is not a join-semilattice]] Directed set is a more general concept than (join) semilattice: every [[Semilattice|join semilattice]] is a directed set, as the join or least upper bound of two elements is the desired <math>c.</math> The converse does not hold however, witness the directed set {1000,0001,1101,1011,1111} [[Coordinatewise order|ordered bitwise]] (e.g. <math>1000 \leq 1011</math> holds, but <math>0001 \leq 1000</math> does not, since in the last bit 1 > 0), where {1000,0001} has three upper bounds but no {{em|least}} upper bound, cf. picture. (Also note that without 1111, the set is not directed.) ==Directed subsets== The order relation in a directed set is not required to be [[Antisymmetric relation|antisymmetric]], and therefore directed sets are not always [[partial order]]s. However, the term {{em|directed set}} is also used frequently in the context of posets. In this setting, a subset <math>A</math> of a partially ordered set <math>(P, \leq)</math> is called a '''directed subset''' if it is a directed set according to the same partial order: in other words, it is not the [[empty set]], and every pair of elements has an upper bound. Here the order relation on the elements of <math>A</math> is inherited from <math>P</math>; for this reason, reflexivity and transitivity need not be required explicitly. A directed subset of a poset is not required to be [[Lower set|downward closed]]; a subset of a poset is directed if and only if its downward closure is an [[Ideal (order theory)|ideal]]. While the definition of a directed set is for an "upward-directed" set (every pair of elements has an upper bound), it is also possible to define a downward-directed set in which every pair of elements has a common lower bound. A subset of a poset is downward-directed if and only if its upper closure is a [[Filter (set theory)|filter]]. Directed subsets are used in [[domain theory]], which studies [[Complete partial order|directed-complete partial order]]s.<ref>Gierz, p. 2.</ref> These are posets in which every upward-directed set is required to have a [[least upper bound]]. In this context, directed subsets again provide a generalization of convergent sequences.{{explain|reason=Again? Convergent sequences are never mentioned in this article.|date=December 2020}} ==See also== * {{annotated link|Centered set}} * {{annotated link|Filtered category}} * {{annotated link|Filters in topology}} * {{annotated link|Linked set}} * {{annotated link|Net (mathematics)}} ==Notes== {{reflist|group=note}} {{reflist}} ==References== * {{Kelley 1975}}{{sfn whitelist|CITEREFKelley1975}} * Gierz, Hofmann, Keimel, ''et al.'' (2003), ''Continuous Lattices and Domains'', Cambridge University Press. {{ISBN|0-521-80338-1}}. {{Order theory}} [[Category:Binary relations]] [[Category:General topology]] [[Category:Order theory]] </textarea><div class="templatesUsed"><div class="mw-templatesUsedExplanation"><p><span id="templatesused">Pages transcluded onto the current version of this page<span class="posteditwindowhelplinks"> (<a href="/wiki/Help:Transclusion" title="Help:Transclusion">help</a>)</span>:</span> </p></div><ul> <li><a href="/wiki/Centered_set" title="Centered set">Centered set</a> (<a href="/w/index.php?title=Centered_set&action=edit" title="Centered set">edit</a>) </li><li><a href="/wiki/Filtered_category" title="Filtered category">Filtered category</a> (<a href="/w/index.php?title=Filtered_category&action=edit" title="Filtered category">edit</a>) </li><li><a href="/wiki/Filters_in_topology" title="Filters in topology">Filters in topology</a> (<a href="/w/index.php?title=Filters_in_topology&action=edit" title="Filters in topology">edit</a>) </li><li><a href="/wiki/Linked_set" title="Linked set">Linked set</a> (<a href="/w/index.php?title=Linked_set&action=edit" title="Linked set">edit</a>) </li><li><a href="/wiki/Net_(mathematics)" title="Net (mathematics)">Net (mathematics)</a> (<a href="/w/index.php?title=Net_(mathematics)&action=edit" title="Net (mathematics)">edit</a>) </li><li><a href="/wiki/Template:Annotated_link" title="Template:Annotated link">Template:Annotated link</a> (<a href="/w/index.php?title=Template:Annotated_link&action=edit" title="Template:Annotated link">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Catalog_lookup_link" title="Template:Catalog lookup link">Template:Catalog lookup link</a> (<a href="/w/index.php?title=Template:Catalog_lookup_link&action=edit" title="Template:Catalog lookup link">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Category_handler" title="Template:Category handler">Template:Category handler</a> (<a href="/w/index.php?title=Template:Category_handler&action=edit" title="Template:Category handler">view source</a>) (protected)</li><li><a href="/wiki/Template:Cite_book" title="Template:Cite book">Template:Cite book</a> (<a href="/w/index.php?title=Template:Cite_book&action=edit" title="Template:Cite book">view source</a>) (protected)</li><li><a href="/wiki/Template:Delink" title="Template:Delink">Template:Delink</a> (<a href="/w/index.php?title=Template:Delink&action=edit" title="Template:Delink">view source</a>) (protected)</li><li><a href="/wiki/Template:Em" title="Template:Em">Template:Em</a> (<a href="/w/index.php?title=Template:Em&action=edit" title="Template:Em">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Explain" title="Template:Explain">Template:Explain</a> (<a href="/w/index.php?title=Template:Explain&action=edit" title="Template:Explain">view source</a>) (extended confirmed protected)</li><li><a href="/wiki/Template:Fix" title="Template:Fix">Template:Fix</a> (<a href="/w/index.php?title=Template:Fix&action=edit" title="Template:Fix">view source</a>) (protected)</li><li><a href="/wiki/Template:Fix/category" title="Template:Fix/category">Template:Fix/category</a> (<a href="/w/index.php?title=Template:Fix/category&action=edit" title="Template:Fix/category">view source</a>) (protected)</li><li><a href="/wiki/Template:Hair_space" title="Template:Hair space">Template:Hair space</a> (<a href="/w/index.php?title=Template:Hair_space&action=edit" title="Template:Hair space">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Hairsp" class="mw-redirect" title="Template:Hairsp">Template:Hairsp</a> (<a href="/w/index.php?title=Template:Hairsp&action=edit" class="mw-redirect" title="Template:Hairsp">view source</a>) (semi-protected)</li><li><a href="/wiki/Template:Hlist/styles.css" title="Template:Hlist/styles.css">Template:Hlist/styles.css</a> (<a href="/w/index.php?title=Template:Hlist/styles.css&action=edit" title="Template:Hlist/styles.css">view source</a>) (protected)</li><li><a href="/wiki/Template:ISBN" title="Template:ISBN">Template:ISBN</a> (<a href="/w/index.php?title=Template:ISBN&action=edit" title="Template:ISBN">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Ifeq" title="Template:Ifeq">Template:Ifeq</a> (<a href="/w/index.php?title=Template:Ifeq&action=edit" title="Template:Ifeq">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Kelley_1975" title="Template:Kelley 1975">Template:Kelley 1975</a> (<a href="/w/index.php?title=Template:Kelley_1975&action=edit" title="Template:Kelley 1975">edit</a>) </li><li><a href="/wiki/Template:Main_other" title="Template:Main other">Template:Main other</a> (<a href="/w/index.php?title=Template:Main_other&action=edit" title="Template:Main other">view source</a>) (protected)</li><li><a href="/wiki/Template:Mvar" title="Template:Mvar">Template:Mvar</a> (<a href="/w/index.php?title=Template:Mvar&action=edit" title="Template:Mvar">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Navbox" title="Template:Navbox">Template:Navbox</a> (<a href="/w/index.php?title=Template:Navbox&action=edit" title="Template:Navbox">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Order_theory" title="Template:Order theory">Template:Order theory</a> (<a href="/w/index.php?title=Template:Order_theory&action=edit" title="Template:Order theory">edit</a>) </li><li><a href="/wiki/Template:P2" title="Template:P2">Template:P2</a> (<a href="/w/index.php?title=Template:P2&action=edit" title="Template:P2">view source</a>) (protected)</li><li><a href="/wiki/Template:Pagetype" title="Template:Pagetype">Template:Pagetype</a> (<a href="/w/index.php?title=Template:Pagetype&action=edit" title="Template:Pagetype">view source</a>) (protected)</li><li><a href="/wiki/Template:Pi" title="Template:Pi">Template:Pi</a> (<a href="/w/index.php?title=Template:Pi&action=edit" title="Template:Pi">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Reflist" title="Template:Reflist">Template:Reflist</a> (<a href="/w/index.php?title=Template:Reflist&action=edit" title="Template:Reflist">view source</a>) (protected)</li><li><a href="/wiki/Template:Reflist/styles.css" title="Template:Reflist/styles.css">Template:Reflist/styles.css</a> (<a href="/w/index.php?title=Template:Reflist/styles.css&action=edit" title="Template:Reflist/styles.css">view source</a>) (protected)</li><li><a href="/wiki/Template:SDcat" title="Template:SDcat">Template:SDcat</a> (<a href="/w/index.php?title=Template:SDcat&action=edit" title="Template:SDcat">view source</a>) (protected)</li><li><a href="/wiki/Template:See_also" title="Template:See also">Template:See also</a> (<a href="/w/index.php?title=Template:See_also&action=edit" title="Template:See also">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Sfn" title="Template:Sfn">Template:Sfn</a> (<a href="/w/index.php?title=Template:Sfn&action=edit" title="Template:Sfn">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Sfn_whitelist" title="Template:Sfn whitelist">Template:Sfn whitelist</a> (<a href="/w/index.php?title=Template:Sfn_whitelist&action=edit" title="Template:Sfn whitelist">view source</a>) (semi-protected)</li><li><a href="/wiki/Template:Short_description" title="Template:Short description">Template:Short description</a> (<a href="/w/index.php?title=Template:Short_description&action=edit" title="Template:Short description">view source</a>) (protected)</li><li><a href="/wiki/Template:Short_description/lowercasecheck" title="Template:Short description/lowercasecheck">Template:Short description/lowercasecheck</a> (<a href="/w/index.php?title=Template:Short_description/lowercasecheck&action=edit" title="Template:Short description/lowercasecheck">view source</a>) (protected)</li><li><a href="/wiki/Template:Visible_anchor" title="Template:Visible anchor">Template:Visible anchor</a> (<a href="/w/index.php?title=Template:Visible_anchor&action=edit" title="Template:Visible anchor">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Visible_anchor/styles.css" title="Template:Visible anchor/styles.css">Template:Visible anchor/styles.css</a> (<a href="/w/index.php?title=Template:Visible_anchor/styles.css&action=edit" title="Template:Visible anchor/styles.css">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Yesno" title="Template:Yesno">Template:Yesno</a> (<a href="/w/index.php?title=Template:Yesno&action=edit" title="Template:Yesno">view source</a>) (protected)</li><li><a href="/wiki/Template:Yesno-no" title="Template:Yesno-no">Template:Yesno-no</a> (<a href="/w/index.php?title=Template:Yesno-no&action=edit" title="Template:Yesno-no">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Yesno-yes" title="Template:Yesno-yes">Template:Yesno-yes</a> (<a href="/w/index.php?title=Template:Yesno-yes&action=edit" title="Template:Yesno-yes">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Annotated_link" title="Module:Annotated link">Module:Annotated link</a> (<a href="/w/index.php?title=Module:Annotated_link&action=edit" title="Module:Annotated link">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Arguments" title="Module:Arguments">Module:Arguments</a> (<a href="/w/index.php?title=Module:Arguments&action=edit" title="Module:Arguments">view source</a>) (protected)</li><li><a href="/wiki/Module:Catalog_lookup_link" title="Module:Catalog lookup link">Module:Catalog lookup link</a> (<a href="/w/index.php?title=Module:Catalog_lookup_link&action=edit" title="Module:Catalog lookup link">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Category_handler" title="Module:Category handler">Module:Category handler</a> (<a href="/w/index.php?title=Module:Category_handler&action=edit" title="Module:Category handler">view source</a>) (protected)</li><li><a href="/wiki/Module:Category_handler/blacklist" title="Module:Category handler/blacklist">Module:Category handler/blacklist</a> (<a href="/w/index.php?title=Module:Category_handler/blacklist&action=edit" title="Module:Category handler/blacklist">view source</a>) (protected)</li><li><a href="/wiki/Module:Category_handler/config" title="Module:Category handler/config">Module:Category handler/config</a> (<a href="/w/index.php?title=Module:Category_handler/config&action=edit" title="Module:Category handler/config">view source</a>) (protected)</li><li><a href="/wiki/Module:Category_handler/data" title="Module:Category handler/data">Module:Category handler/data</a> (<a href="/w/index.php?title=Module:Category_handler/data&action=edit" title="Module:Category handler/data">view source</a>) (protected)</li><li><a href="/wiki/Module:Category_handler/shared" title="Module:Category handler/shared">Module:Category handler/shared</a> (<a href="/w/index.php?title=Module:Category_handler/shared&action=edit" title="Module:Category handler/shared">view source</a>) (protected)</li><li><a href="/wiki/Module:Check_for_unknown_parameters" title="Module:Check for unknown parameters">Module:Check for unknown parameters</a> (<a href="/w/index.php?title=Module:Check_for_unknown_parameters&action=edit" title="Module:Check for unknown parameters">view source</a>) (protected)</li><li><a href="/wiki/Module:Check_isxn" title="Module:Check isxn">Module:Check isxn</a> (<a href="/w/index.php?title=Module:Check_isxn&action=edit" title="Module:Check isxn">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Citation/CS1" title="Module:Citation/CS1">Module:Citation/CS1</a> (<a href="/w/index.php?title=Module:Citation/CS1&action=edit" title="Module:Citation/CS1">view source</a>) (protected)</li><li><a href="/wiki/Module:Citation/CS1/COinS" title="Module:Citation/CS1/COinS">Module:Citation/CS1/COinS</a> (<a href="/w/index.php?title=Module:Citation/CS1/COinS&action=edit" title="Module:Citation/CS1/COinS">view source</a>) (protected)</li><li><a href="/wiki/Module:Citation/CS1/Configuration" title="Module:Citation/CS1/Configuration">Module:Citation/CS1/Configuration</a> (<a href="/w/index.php?title=Module:Citation/CS1/Configuration&action=edit" title="Module:Citation/CS1/Configuration">view source</a>) (protected)</li><li><a href="/wiki/Module:Citation/CS1/Date_validation" title="Module:Citation/CS1/Date validation">Module:Citation/CS1/Date validation</a> (<a href="/w/index.php?title=Module:Citation/CS1/Date_validation&action=edit" title="Module:Citation/CS1/Date validation">view source</a>) (protected)</li><li><a href="/wiki/Module:Citation/CS1/Identifiers" title="Module:Citation/CS1/Identifiers">Module:Citation/CS1/Identifiers</a> (<a href="/w/index.php?title=Module:Citation/CS1/Identifiers&action=edit" title="Module:Citation/CS1/Identifiers">view source</a>) (protected)</li><li><a href="/wiki/Module:Citation/CS1/Utilities" title="Module:Citation/CS1/Utilities">Module:Citation/CS1/Utilities</a> (<a href="/w/index.php?title=Module:Citation/CS1/Utilities&action=edit" title="Module:Citation/CS1/Utilities">view source</a>) (protected)</li><li><a href="/wiki/Module:Citation/CS1/Whitelist" title="Module:Citation/CS1/Whitelist">Module:Citation/CS1/Whitelist</a> (<a href="/w/index.php?title=Module:Citation/CS1/Whitelist&action=edit" title="Module:Citation/CS1/Whitelist">view source</a>) (protected)</li><li><a href="/wiki/Module:Citation/CS1/styles.css" title="Module:Citation/CS1/styles.css">Module:Citation/CS1/styles.css</a> (<a href="/w/index.php?title=Module:Citation/CS1/styles.css&action=edit" title="Module:Citation/CS1/styles.css">view source</a>) (protected)</li><li><a href="/wiki/Module:Delink" title="Module:Delink">Module:Delink</a> (<a href="/w/index.php?title=Module:Delink&action=edit" title="Module:Delink">view source</a>) (protected)</li><li><a href="/wiki/Module:Disambiguation/templates" title="Module:Disambiguation/templates">Module:Disambiguation/templates</a> (<a href="/w/index.php?title=Module:Disambiguation/templates&action=edit" title="Module:Disambiguation/templates">view source</a>) (protected)</li><li><a href="/wiki/Module:Footnotes" title="Module:Footnotes">Module:Footnotes</a> (<a href="/w/index.php?title=Module:Footnotes&action=edit" title="Module:Footnotes">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Footnotes/anchor_id_list" title="Module:Footnotes/anchor id list">Module:Footnotes/anchor id list</a> (<a href="/w/index.php?title=Module:Footnotes/anchor_id_list&action=edit" title="Module:Footnotes/anchor id list">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Footnotes/anchor_id_list/data" title="Module:Footnotes/anchor id list/data">Module:Footnotes/anchor id list/data</a> (<a href="/w/index.php?title=Module:Footnotes/anchor_id_list/data&action=edit" title="Module:Footnotes/anchor id list/data">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Footnotes/whitelist" title="Module:Footnotes/whitelist">Module:Footnotes/whitelist</a> (<a href="/w/index.php?title=Module:Footnotes/whitelist&action=edit" title="Module:Footnotes/whitelist">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Format_link" title="Module:Format link">Module:Format link</a> (<a href="/w/index.php?title=Module:Format_link&action=edit" title="Module:Format link">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:GetShortDescription" title="Module:GetShortDescription">Module:GetShortDescription</a> (<a href="/w/index.php?title=Module:GetShortDescription&action=edit" title="Module:GetShortDescription">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Hatnote" title="Module:Hatnote">Module:Hatnote</a> (<a href="/w/index.php?title=Module:Hatnote&action=edit" title="Module:Hatnote">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Hatnote/styles.css" title="Module:Hatnote/styles.css">Module:Hatnote/styles.css</a> (<a href="/w/index.php?title=Module:Hatnote/styles.css&action=edit" title="Module:Hatnote/styles.css">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Hatnote_list" title="Module:Hatnote list">Module:Hatnote list</a> (<a href="/w/index.php?title=Module:Hatnote_list&action=edit" title="Module:Hatnote list">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Labelled_list_hatnote" title="Module:Labelled list hatnote">Module:Labelled list hatnote</a> (<a href="/w/index.php?title=Module:Labelled_list_hatnote&action=edit" title="Module:Labelled list hatnote">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Lang" title="Module:Lang">Module:Lang</a> (<a href="/w/index.php?title=Module:Lang&action=edit" title="Module:Lang">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Lang/ISO_639_synonyms" title="Module:Lang/ISO 639 synonyms">Module:Lang/ISO 639 synonyms</a> (<a href="/w/index.php?title=Module:Lang/ISO_639_synonyms&action=edit" title="Module:Lang/ISO 639 synonyms">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Lang/data" title="Module:Lang/data">Module:Lang/data</a> (<a href="/w/index.php?title=Module:Lang/data&action=edit" title="Module:Lang/data">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Lang/data/iana_languages" title="Module:Lang/data/iana languages">Module:Lang/data/iana languages</a> (<a href="/w/index.php?title=Module:Lang/data/iana_languages&action=edit" title="Module:Lang/data/iana languages">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Lang/data/iana_regions" title="Module:Lang/data/iana regions">Module:Lang/data/iana regions</a> (<a href="/w/index.php?title=Module:Lang/data/iana_regions&action=edit" title="Module:Lang/data/iana regions">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Lang/data/iana_scripts" title="Module:Lang/data/iana scripts">Module:Lang/data/iana scripts</a> (<a href="/w/index.php?title=Module:Lang/data/iana_scripts&action=edit" title="Module:Lang/data/iana scripts">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Lang/data/iana_suppressed_scripts" title="Module:Lang/data/iana suppressed scripts">Module:Lang/data/iana suppressed scripts</a> (<a href="/w/index.php?title=Module:Lang/data/iana_suppressed_scripts&action=edit" title="Module:Lang/data/iana suppressed scripts">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Lang/data/iana_variants" title="Module:Lang/data/iana variants">Module:Lang/data/iana variants</a> (<a href="/w/index.php?title=Module:Lang/data/iana_variants&action=edit" title="Module:Lang/data/iana variants">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Namespace_detect/config" title="Module:Namespace detect/config">Module:Namespace detect/config</a> (<a href="/w/index.php?title=Module:Namespace_detect/config&action=edit" title="Module:Namespace detect/config">view source</a>) (protected)</li><li><a href="/wiki/Module:Namespace_detect/data" title="Module:Namespace detect/data">Module:Namespace detect/data</a> (<a href="/w/index.php?title=Module:Namespace_detect/data&action=edit" title="Module:Namespace detect/data">view source</a>) (protected)</li><li><a href="/wiki/Module:Navbar" title="Module:Navbar">Module:Navbar</a> (<a href="/w/index.php?title=Module:Navbar&action=edit" title="Module:Navbar">view source</a>) (protected)</li><li><a href="/wiki/Module:Navbar/configuration" title="Module:Navbar/configuration">Module:Navbar/configuration</a> (<a href="/w/index.php?title=Module:Navbar/configuration&action=edit" title="Module:Navbar/configuration">view source</a>) (protected)</li><li><a href="/wiki/Module:Navbar/styles.css" title="Module:Navbar/styles.css">Module:Navbar/styles.css</a> (<a href="/w/index.php?title=Module:Navbar/styles.css&action=edit" title="Module:Navbar/styles.css">view source</a>) (protected)</li><li><a href="/wiki/Module:Navbox" title="Module:Navbox">Module:Navbox</a> (<a href="/w/index.php?title=Module:Navbox&action=edit" title="Module:Navbox">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Navbox/configuration" title="Module:Navbox/configuration">Module:Navbox/configuration</a> (<a href="/w/index.php?title=Module:Navbox/configuration&action=edit" title="Module:Navbox/configuration">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Navbox/styles.css" title="Module:Navbox/styles.css">Module:Navbox/styles.css</a> (<a href="/w/index.php?title=Module:Navbox/styles.css&action=edit" title="Module:Navbox/styles.css">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Pagetype" title="Module:Pagetype">Module:Pagetype</a> (<a href="/w/index.php?title=Module:Pagetype&action=edit" title="Module:Pagetype">view source</a>) (protected)</li><li><a href="/wiki/Module:Pagetype/config" title="Module:Pagetype/config">Module:Pagetype/config</a> (<a href="/w/index.php?title=Module:Pagetype/config&action=edit" title="Module:Pagetype/config">view source</a>) (protected)</li><li><a href="/wiki/Module:Pagetype/disambiguation" class="mw-redirect" title="Module:Pagetype/disambiguation">Module:Pagetype/disambiguation</a> (<a href="/w/index.php?title=Module:Pagetype/disambiguation&action=edit" class="mw-redirect" title="Module:Pagetype/disambiguation">view source</a>) (protected)</li><li><a href="/wiki/Module:Pagetype/rfd" title="Module:Pagetype/rfd">Module:Pagetype/rfd</a> (<a href="/w/index.php?title=Module:Pagetype/rfd&action=edit" title="Module:Pagetype/rfd">view source</a>) (protected)</li><li><a href="/wiki/Module:Pagetype/setindex" title="Module:Pagetype/setindex">Module:Pagetype/setindex</a> (<a href="/w/index.php?title=Module:Pagetype/setindex&action=edit" title="Module:Pagetype/setindex">view source</a>) (protected)</li><li><a href="/wiki/Module:Pagetype/softredirect" title="Module:Pagetype/softredirect">Module:Pagetype/softredirect</a> (<a href="/w/index.php?title=Module:Pagetype/softredirect&action=edit" title="Module:Pagetype/softredirect">view source</a>) (protected)</li><li><a href="/wiki/Module:SDcat" title="Module:SDcat">Module:SDcat</a> (<a href="/w/index.php?title=Module:SDcat&action=edit" title="Module:SDcat">view source</a>) (protected)</li><li><a href="/wiki/Module:String" title="Module:String">Module:String</a> (<a href="/w/index.php?title=Module:String&action=edit" title="Module:String">view source</a>) (protected)</li><li><a href="/wiki/Module:TableTools" title="Module:TableTools">Module:TableTools</a> (<a href="/w/index.php?title=Module:TableTools&action=edit" title="Module:TableTools">view source</a>) (protected)</li><li><a href="/wiki/Module:Unicode_data" title="Module:Unicode data">Module:Unicode data</a> (<a href="/w/index.php?title=Module:Unicode_data&action=edit" title="Module:Unicode data">view source</a>) (template editor protected)</li><li><a href="/wiki/Module:Unsubst" title="Module:Unsubst">Module:Unsubst</a> (<a href="/w/index.php?title=Module:Unsubst&action=edit" title="Module:Unsubst">view source</a>) (protected)</li><li><a href="/wiki/Module:Wikitext_Parsing" title="Module:Wikitext Parsing">Module:Wikitext Parsing</a> (<a href="/w/index.php?title=Module:Wikitext_Parsing&action=edit" title="Module:Wikitext Parsing">view source</a>) (protected)</li><li><a href="/wiki/Module:Yesno" title="Module:Yesno">Module:Yesno</a> (<a href="/w/index.php?title=Module:Yesno&action=edit" title="Module:Yesno">view source</a>) (protected)</li></ul></div><p id="mw-returnto">Return to <a href="/wiki/Directed_set" title="Directed set">Directed set</a>.</p> <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="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/wiki/Directed_set">https://en.wikipedia.org/wiki/Directed_set</a>"</div></div> <div id="catlinks" class="catlinks catlinks-allhidden" data-mw="interface"></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Directed_set&action=edit&undo=1193248815&undoafter=1176830014&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-9j6r5","wgBackendResponseTime":549,"wgPageParseReport":{"limitreport":{"cputime":"0.073","walltime":"0.095","ppvisitednodes":{"value":418,"limit":1000000},"postexpandincludesize":{"value":17740,"limit":2097152},"templateargumentsize":{"value":6556,"limit":2097152},"expansiondepth":{"value":9,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":469,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 64.015 1 -total"," 99.86% 63.923 2 Template:Blocked_text"," 39.74% 25.440 1 Template:Colocationwebhost"," 38.81% 24.843 2 Template:Replace"," 34.98% 22.393 1 Template:Hidden"," 15.40% 9.860 1 Template:Tlx"," 13.20% 8.452 1 Template:Hidden_begin"," 2.55% 1.635 1 Template:Hidden_end"," 2.30% 1.471 1 MediaWiki:Wikimedia-globalblocking-blockedtext-mistake"," 1.78% 1.142 1 MediaWiki:Wikimedia-globalblocking-blockedtext-mistake-email-steward"]},"scribunto":{"limitreport-timeusage":{"value":"0.014","limit":"10.000"},"limitreport-memusage":{"value":1043292,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-9j6r5","timestamp":"20241124130047","ttl":2592000,"transientcontent":false}}});});</script> </body> </html>