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@media (max-width:350px) { .adslot_1 { width: 300px; height: 250px; } }</style>  <ins class="adsbygoogle adslot_1"data-ad-client=ca-pub-5629769610382312 data-ad-slot=9569618821 style=display:inline-block;> </ins> <script async src=//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js></script> <script>(adsbygoogle = window.adsbygoogle || []).push({});</script> </center></div><p align=justify>In other words, a space $$X$$ is connected if it is not the union of two non-empty disjoint open sets.<p><strong> Example: </strong><p align=justify>Every indiscrete space is connected.<p align=justify>Let $$X$$ be an indiscrete space, then $$X$$ is the only non-empty open set, so we cannot find the disconnection of $$X$$. Hence $$X$$ is connected.<p> <p><strong> Connected Subspace </strong><div style="margin: 15px 0 15px 0;"><script async src=//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js></script><ins class=adsbygoogle data-ad-client=ca-pub-5629769610382312 data-ad-format=auto data-ad-slot=2451247916 data-full-width-responsive=true style=display:block> </ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script></div><p align=justify>A subspace $$Y$$ of a topological space is said to be a connected subspace if $$Y$$ is connected as a topological space in its own right.<p><strong> Example: </strong><p align=justify>The subset of a connected space may not be connected.<p align=justify>The set of real numbers $$\mathbb{R}$$ cannot be represented as the union of two disjoint non-empty sets, so $$\mathbb{R}$$ is a connected space. <br> Next suppose that $$A = \mathbb{Q} \cap \left] { – \infty ,\sqrt 2 } \right[$$ and $$B = \mathbb{Q} \cap \left] {\sqrt 2 ,\infty } \right[$$ <br> Since $$A = \mathbb{Q} \cap \left] { – \infty ,\sqrt 2 } \right[$$ and $$B = \mathbb{Q} \cap \left] {\sqrt 2 ,\infty } \right[$$ are open subsets of $$\mathbb{R}$$, so $$A$$ and $$B$$ are open subsets of $$\mathbb{Q}$$. Also $$A \cap B = \phi $$ and $$A \cup B = \mathbb{Q}$$. This shows that $$\mathbb{Q}$$ is a disconnected subspace of $$\mathbb{R}$$.<p align=justify><p align=justify><strong> Theorems </strong> <br> • A topological space is connected if and only if it cannot be represented as the union of two disjoint non-empty closed sets. <br> • An infinite set with co-finite topology is a connected space. <br> • Any continuous image of a connected space is connected. <br> • The range of a continuous real unction defined on a connected space is an interval. <br> • If $$\left\{ {A,B} \right\}$$ is a disconnection of $$X$$ and $$C$$ is a connected subspace of $$X$$, then $$B$$ is contained either in $$A$$ or in $$B$$.<p align=justify><p align=justify><strong> Characterization of a Connected Space </strong><div style="margin: 15px 0;"><center>  <div id=article2><script>googletag.cmd.push(function() { googletag.display('article2'); });</script></div> </center></div><p>In a space, the following are equivalent: <br> • $$X$$is connected. <br> • The only open and closed subsets of $$X$$ are $$\phi $$, $$X$$. <br> • There does not exist a continuous map $$f:X \to \left\{ {0,1} \right\}$$ from a space $$X$$ onto the discrete space $$\left\{ {0,1} \right\}$$.<div class="themify_builder_content themify_builder_content-117 themify_builder themify_builder_front"data-postid=117 id=themify_builder_content-117></div></div></div></article><div class="post-nav clearfix"><span class=prev> <a href=https://www.emathzone.com/tutorials/general-topology/disconnected-space.html rel=next> <span class=arrow> ⇐ </span> Disconnected Space </a> </span><span class=next> <a href=https://www.emathzone.com/tutorials/general-topology/components-of-a-space.html rel=prev> <span class=arrow> ⇒ </span> Components of a Space ⇒ </a> </span></div><div class=commentwrap id=comments><div class=comment-respond id=respond><h3 class=comment-reply-title id=reply-title>Leave a Reply <small> <a href=/tutorials/general-topology/connected-space.html#respond id=cancel-comment-reply-link rel=nofollow style=display:none;> Cancel reply </a> </small></h3><form action=https://www.emathzone.com/wp-comments-post.php class=comment-form id=commentform method=post><p class=comment-notes><span id=email-notes> Your email address will not be published. </span> Required fields are marked <span class=required> * </span><p class=comment-form-comment><label for=comment>Comment</label> <textarea cols=45 id=comment maxlength=65525 name=comment required rows=8></textarea><p class=comment-form-author><label for=author>Name <span class=required> * </span></label> <input id=author maxlength=245 name=author required size=30><p class=comment-form-email><label for=email>Email <span class=required> * </span></label> <input aria-describedby=email-notes id=email maxlength=100 name=email required size=30><p class=comment-form-url><label for=url>Website</label> <input id=url maxlength=200 name=url size=30><p class=form-submit><input value="Post Comment"class=submit id=submit name=submit type=submit> <input id=comment_post_ID name=comment_post_ID type=hidden value=117> <input id=comment_parent name=comment_parent type=hidden value=0><p style="display: none;"><input id=akismet_comment_nonce name=akismet_comment_nonce type=hidden value=de2fe4340f></p><input id=ak_js name=ak_js type=hidden value=38><textarea style="display: none !important;"cols=45 maxlength=100 name=ak_hp_textarea rows=8></textarea></form></div></div><div style="margin: 15px 0;"><center>  <div id=article1><script>googletag.cmd.push(function() { googletag.display('article1'); 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