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href="../../../Definitions/Rational-Cauchy-Sequences" data-for="Definitions/Rational-Cauchy-Sequences">Rational Cauchy Sequences</a></li><li><a href="../../../Definitions/Subcover" data-for="Definitions/Subcover">Subcover</a></li><li><a href="../../../Definitions/Subnet" data-for="Definitions/Subnet">Subnet</a></li></ul></div></li><li><div class="folder-container"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="12" viewBox="5 8 14 8" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="folder-icon"><polyline points="6 9 12 15 18 9"></polyline></svg><div data-folderpath="Lectures"><button class="folder-button"><span class="folder-title">Lectures</span></button></div></div><div class="folder-outer "><ul style="padding-left:1.4rem;" class="content" data-folderul="Lectures"><li><a href="../../../Lectures/Lecture-1---1.1-Sets-and-Numbers" data-for="Lectures/Lecture-1---1.1-Sets-and-Numbers">Lecture 1 - 1.1 Sets and Numbers</a></li><li><a href="../../../Lectures/Lecture-2" data-for="Lectures/Lecture-2">Lecture 2</a></li><li><a href="../../../Lectures/Lecture-3" data-for="Lectures/Lecture-3">Lecture 3</a></li><li><a href="../../../Lectures/Lecture-4---1.2-Metric-Spaces" data-for="Lectures/Lecture-4---1.2-Metric-Spaces">Lecture 4 - 1.2 Metric Spaces</a></li><li><a href="../../../Lectures/Lecture-5" data-for="Lectures/Lecture-5">Lecture 5</a></li><li><a href="../../../Lectures/Lecture-6---2.1-Topology" data-for="Lectures/Lecture-6---2.1-Topology">Lecture 6 - 2.1 Topology</a></li><li><a href="../../../Lectures/Lecture-7" data-for="Lectures/Lecture-7">Lecture 7</a></li><li><a href="../../../Lectures/Lecture-8" data-for="Lectures/Lecture-8">Lecture 8</a></li><li><a href="../../../Lectures/Lecture-11" data-for="Lectures/Lecture-11">Lecture 11</a></li><li><a href="../../../Lectures/Lecture-12---Induced-Topologies" data-for="Lectures/Lecture-12---Induced-Topologies">Lecture 12 - Induced Topologies</a></li><li><a href="../../../Lectures/Lecture-13---Measure-Theory" data-for="Lectures/Lecture-13---Measure-Theory">Lecture 13 - Measure Theory</a></li></ul></div></li></ul></div></li><li id="explorer-end"></li></ul></div></div></div><div class="center"><div class="page-header"><div class="popover-hint"><nav class="breadcrumb-container" aria-label="breadcrumbs"><div class="breadcrumb-element"><a href="../../../">Home</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/">Definitions</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/">Topological Spaces</a><p> ❯ </p></div><div class="breadcrumb-element"><a href="../../../Definitions/Topological-Spaces/Terminologies/">Terminologies</a><p> ❯ </p></div><div class="breadcrumb-element"><a href>Connected</a></div></nav><h1 class="article-title">Connected</h1><p show-comma="true" class="content-meta"><time datetime="2025-03-01T15:59:38.422Z">01 Mar 2025</time><span>1 min read</span></p></div></div><article class="popover-hint"><h1 id="definition">Definition<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#definition" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
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<p>A <a href="../../../Definitions/Topological-Spaces/Topological-Space" class="internal alias" data-slug="Definitions/Topological-Spaces/Topological-Space">topological space</a> is <strong>connected</strong> if it is not a union of two non-empty <a href="../../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open sets</a>.</p>
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<p>i.e. if you draw the two non-empty <a href="../../../Definitions/Sets/Open-Sets" class="internal alias" data-slug="Definitions/Sets/Open-Sets">open sets</a> on the graph, if you have to lift your pen, it will not be connected.</p>
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<h1 id="examples">Examples<a role="anchor" aria-hidden="true" tabindex="-1" data-no-popover="true" href="#examples" class="internal"><svg width="18" height="18" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></a></h1>
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<p>Not connected:<br/>
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<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">5</span><span class="mclose">⟩</span></span></span></span><br/>
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Connected:<br/>
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<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0.5</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">5</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⟨</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">5</span><span class="mclose">⟩</span></span></span></span></p></article><hr/><div class="page-footer"></div></div><div class="right sidebar"><div class="graph"><h3>Graph View</h3><div class="graph-outer"><div id="graph-container" data-cfg="{"drag":true,"zoom":true,"depth":1,"scale":1.1,"repelForce":0.5,"centerForce":0.3,"linkDistance":30,"fontSize":0.6,"opacityScale":1,"showTags":true,"removeTags":[],"focusOnHover":false,"enableRadial":false}"></div><button id="global-graph-icon" aria-label="Global Graph"><svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 55 55" fill="currentColor" xml:space="preserve"><path d="M49,0c-3.309,0-6,2.691-6,6c0,1.035,0.263,2.009,0.726,2.86l-9.829,9.829C32.542,17.634,30.846,17,29,17
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