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/*]]>*/</style><body class=ns-0><div id=globalWrapper><div align=left id=header-logo><table border=0 cellpadding=0 cellspacing=0 width=100%><tbody><tr bgcolor=#02F5FB><td><a title="Back to main page"bgcolor=#02F5FB href=index.php> <img src=http://www.academickids.com/encyclopedia/encyclopedia.jpg> </a></table><table border=0 cellpadding=10 cellspacing=0 width=100%><tbody><tr align=center><td><script async src=https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js></script>  <ins class=adsbygoogle data-ad-client=ca-pub-6046499734384137 data-ad-format=auto data-ad-slot=3599769782 data-full-width-responsive=true style=display:block> </ins> <script>(adsbygoogle = window.adsbygoogle || []).push({});</script> <br></table></div><div class=spacer150px></div><div class=portlet id=p-reklam><h5></h5><script async src=https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js></script><ins class=adsbygoogle data-ad-client=ca-pub-6046499734384137 data-ad-format=auto data-ad-slot=2550412007 data-full-width-responsive=true style=display:block> </ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script><br><script async src=https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js></script><ins class=adsbygoogle data-ad-client=ca-pub-6046499734384137 data-ad-format=auto data-ad-slot=2550412007 data-full-width-responsive=true style=display:block> </ins><script>(adsbygoogle = window.adsbygoogle || []).push({});</script></div><div id=column-content><div id=content><a id=contentTop name=top> </a><h1 class=firstHeading>Connected space</h1><div id=bodyContent><h3 id=siteSub>From Academic Kids</h3><table align=right border=0 cellpadding=5 cellspacing=5 width=2%><tbody><tr><td align=left valign=bottom><div style="margin-top: 1px"></div></table><div id=contentSub>(Redirected from <a href=/encyclopedia/index.php?title=Connectedness&redirect=no title=Connectedness> Connectedness </a> )</div><p>In <a href=/encyclopedia/index.php/Topology title=Topology> topology </a> and related branches of <a href=/encyclopedia/index.php/Mathematics title=Mathematics> mathematics </a> , a <a title="Topological space"href=/encyclopedia/index.php/Topological_space> topological space </a> <i> X </i> is said to be <b> disconnected </b> if it is the <a title="Union (set theory)"href=/encyclopedia/index.php/Union_%28set_theory%29> union </a> of two <a href=/encyclopedia/index.php/Disjoint title=Disjoint> disjoint </a> <a href=/encyclopedia/index.php/Nonempty title=Nonempty> nonempty </a> <a title="Open set"href=/encyclopedia/index.php/Open_set> open sets </a> . Otherwise, <i> X </i> is said to be <b> connected </b> . A <a href=/encyclopedia/index.php/Subset title=Subset> subset </a> of a topological space is said to be connected if it is connected under its <a title="Subspace (topology)"href=/encyclopedia/index.php/Subspace_%28topology%29> subspace topology </a> . Some authors specifically exclude the <a title="Empty set"href=/encyclopedia/index.php/Empty_set> empty set </a> with its unique topology as a connected space, but this encyclopedia does not follow that practice.<p>For a topological space <i> X </i> the following conditions are equivalent:<ol><li><i> X </i> is connected.<li><i> X </i> cannot be divided into two disjoint nonempty <a title="Closed set"href=/encyclopedia/index.php/Closed_set> closed sets </a> (This follows since the <a title="Complement (set theory)"href=/encyclopedia/index.php/Complement_%28set_theory%29> complement </a> of an open set is closed).<li>The only sets which are both open and closed ( <a title="Clopen set"href=/encyclopedia/index.php/Clopen_set> clopen sets </a> ) are <i> X </i> and the empty set.<li>The only sets with empty <a title="Boundary (topology)"href=/encyclopedia/index.php/Boundary_%28topology%29> boundary </a> are <i> X </i> and the empty set.<li><i> X </i> cannot be written as the union of two nonempty <a title="Separated sets"href=/encyclopedia/index.php/Separated_sets> separated sets </a> .</ol><p>The <a title="Maximal element"href=/encyclopedia/index.php/Maximal_element> maximal </a> nonempty connected subsets of any topological space are called the <b> connected components </b> of the space. The components form a <a title="Partition of a set"href=/encyclopedia/index.php/Partition_of_a_set> partition </a> of the space (that is, they are <a href=/encyclopedia/index.php/Disjoint title=Disjoint> disjoint </a> and their union is the whole space). Every component is a <a title="Closed subset"href=/encyclopedia/index.php/Closed_subset> closed subset </a> of the original space. The components in general need not be open: the components of the <a title="Rational number"href=/encyclopedia/index.php/Rational_number> rational numbers </a> , for instance, are the one-point sets. A space in which all components are one-point sets is called <b> totally disconnected </b> . A space <i> X </i> is totally disconnected iff, for any two elements <i> x </i> and <i> y </i> of <i> X </i> , there exist disjoint <a title="Neighborhood (topology)"href=/encyclopedia/index.php/Neighborhood_%28topology%29> open neighborhoods </a> <i> U </i> of <i> x </i> and <i> V </i> of <i> y </i> such that <i> X </i> is the union of <i> U </i> and <i> V </i> .<table border=0 id=toc><tbody><tr id=toctitle><td align=center><b> Contents </b> <script>showTocToggle("show","hide")</script><tr id=tocinside><td><div class=tocline><a href=#Examples> 1 Examples </a><br></div> <div class=tocline><a href=#Path_connectedness> 2 Path connectedness </a><br></div> <div class=tocline><a href=#Local_connectedness> 3 Local connectedness </a><br></div> <div class=tocline><a href=#Theorems> 4 Theorems </a><br></div> <div class=tocline><a href=#See_also> 5 See also </a><br></div></table><a name=Examples> </a><h2>Examples</h2><ul><li>The space of <a title="Real number"href=/encyclopedia/index.php/Real_number> real numbers </a> with the usual topology is connected.<li>Every <a title="Discrete topological space"class=new href=/encyclopedia/index.php?title=Discrete_topological_space&action=edit> discrete topological space </a> is totally disconnected.<li>The <a title="Cantor set"href=/encyclopedia/index.php/Cantor_set> Cantor set </a> is totally disconnected.</ul><a name=Path_connectedness> </a><h2>Path connectedness</h2><p>The space <i> X </i> is said to be <b> path-connected </b> if for any two points <i> x </i> and <i> y </i> in <i> X </i> there exists a <a href=/encyclopedia/index.php/Continuity title=Continuity> continuous </a> <a title="Function (mathematics)"href=/encyclopedia/index.php/Function_%28mathematics%29> function </a> <i> f </i> from the <a title="Unit interval"href=/encyclopedia/index.php/Unit_interval> unit interval </a> [0,1] to <i> X </i> with <i> f </i> (0) = <i> x </i> and <i> f </i> (1) = <i> y </i> . (This function is called a <i> <a title="Path (topology)"class=new href=/encyclopedia/index.php?title=Path_%28topology%29&action=edit> path </a> </i> from <i> x </i> to <i> y </i> .)<p>Every path-connected space is connected. Example of connected spaces that are not path-connected include the extended <a title="Long line (topology)"href=/encyclopedia/index.php/Long_line_%28topology%29> long line </a> <i> L </i> * and the <i> <a title="Topologist's sine curve"href=/encyclopedia/index.php/Topologist%27s_sine_curve> topologist's sine curve </a> </i> .<p>However, subsets of the <a title="Real line"href=/encyclopedia/index.php/Real_line> real line </a> <b> R </b> are connected <a title="If and only if"href=/encyclopedia/index.php/If_and_only_if> if and only if </a> they are path-connected; these subsets are the <a title="Interval (mathematics)"href=/encyclopedia/index.php/Interval_%28mathematics%29> intervals </a> of <b> R </b> . Also, <a title="Open subset"href=/encyclopedia/index.php/Open_subset> open subsets </a> of <b> R </b> <sup> <i> n </i> </sup> or <b> C </b> <sup> <i> n </i> </sup> are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for <a href=/encyclopedia/index.php/Finite title=Finite> finite </a> topological spaces.<p>A space <i> X </i> is said to be <b> arc-connected </b> if any two distinct points can be joined by an <i> arc </i> , that is a path <i> f </i> which is a <a href=/encyclopedia/index.php/Homeomorphism title=Homeomorphism> homeomorphism </a> between the unit interval [0,1] and its image <i> f </i> ([0,1]). It can be shown any <a title="Hausdorff space"href=/encyclopedia/index.php/Hausdorff_space> Hausdorff space </a> which is path-connected is also arc-connected. An example of a space which is path-connected but not arc-connected is provided by adding a second copy 0' of 0 to the nonnegative real numbers [0,∞). One endows this set with a <a title="Partially ordered set"href=/encyclopedia/index.php/Partially_ordered_set> partial order </a> by specifying that 0'< <i> a </i> for any positive number <i> a </i> , but leaving 0 and 0' incomparable. One then endows this set with the <i> order topology </i> , that is one takes the open intervals (a,b)={x | a&LTx&LTb} and the half-open intervals [0,a)={x | 0≤x&LTa}, [0',a)={x | 0'≤x&LTa} as a <a title="Base (topology)"href=/encyclopedia/index.php/Base_%28topology%29> base </a> for the topology. The resulting space is a <a title="T1 space"href=/encyclopedia/index.php/T1_space> T <sub> 1 </sub> </a> space but not a <a title="Hausdorff space"href=/encyclopedia/index.php/Hausdorff_space> Hausdorff space </a> . Clearly 0 and 0' can be connected by a path but not by an arc in this space.</p><a name=Local_connectedness> </a><h2>Local connectedness</h2><p>A topological space is said to be <b> locally connected </b> if it has a <a title="Base (topology)"href=/encyclopedia/index.php/Base_%28topology%29> base </a> of connected sets. It can be shown that a space <i> X </i> is locally connected if and only if every component of every open set of <i> X </i> is open. The topologist's sine curve shown above is an example of a connected space that is not locally connected.<p>Similarly, a topological space is said to be <b> locally path-connected </b> if it has a base of path-connected sets. An open subset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlier statement about <b> R </b> <sup> <i> n </i> </sup> and <b> C </b> <sup> <i> n </i> </sup> , each of which is locally path-connected. More generally, any <a title="Topological manifold"href=/encyclopedia/index.php/Topological_manifold> topological manifold </a> is locally path-connected.</p><a name=Theorems> </a><h2>Theorems</h2><ul><li><i> Main theorem </i> : Let <i> X </i> and <i> Y </i> be topological spaces and let <i> f </i>  : <i> X </i> → <i> Y </i> be a <a title="Continuous (topology)"href=/encyclopedia/index.php/Continuous_%28topology%29> continuous function </a> . If <i> X </i> is connected (resp. path-connected) then the <a title="Image (mathematics)"href=/encyclopedia/index.php/Image_%28mathematics%29> image </a> <i> f </i> ( <i> X </i> ) is connected (resp. path-connected). The <a title="Intermediate value theorem"href=/encyclopedia/index.php/Intermediate_value_theorem> intermediate value theorem </a> can be considered as a special case of this result.<li>Every path-connected space is connected.<li>Every locally path-connected space is locally connected.<li>A locally path-connected space is path-connected <a href=/encyclopedia/index.php/Iff title=Iff> iff </a> it is connected.<li>The connected components of a space are disjoint unions of the path-connected components.<li>The components of a locally connected space are open (and closed).<li>The <a title="Closure (topology)"href=/encyclopedia/index.php/Closure_%28topology%29> closure </a> of a connected subset is connected.<li>Every <a title="Quotient space"href=/encyclopedia/index.php/Quotient_space> quotient </a> of a connected (resp. path-connected) space is connected (resp. path-connected).<li>Every <a title="Product topology"href=/encyclopedia/index.php/Product_topology> product </a> of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).<li>Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locally path-connected).<li>Every <a href=/encyclopedia/index.php/Manifold title=Manifold> manifold </a> is locally path-connected.</ul><a name=See_also> </a><h2>See also</h2><ul><li><a title="Uniformly connected space"class=new href=/encyclopedia/index.php?title=Uniformly_connected_space&action=edit> uniformly connected space </a><li><a title="Connected component (graph theory)"href=/encyclopedia/index.php/Connected_component_%28graph_theory%29> Connected component (graph theory) </a><li><a title="Separated sets"href=/encyclopedia/index.php/Separated_sets> Separated sets </a><li><a title="Simply connected"href=/encyclopedia/index.php/Simply_connected> Simply connected </a> <a title="Pl:PrzestrzeÅ spÃ³jna"class=new href=/encyclopedia/index.php?title=Pl:Przestrze%C5%84_sp%C3%B3jna&action=edit> pl:PrzestrzeÅ spÃ³jna </a></ul><p><a class=new href=/encyclopedia/index.php?title=Zh:%E8%BF%9E%E9%80%9A%E7%A9%BA%E9%97%B4&action=edit title=Zh:è¿éç©ºé´> zh:连通空间 </a> <a title="De:Zusammenhang (Topologie)"class=new href=/encyclopedia/index.php?title=De:Zusammenhang_%28Topologie%29&action=edit> de:Zusammenhang (Topologie) </a> <a class=new href=/encyclopedia/index.php?title=Fr:Connexit%E9&action=edit> fr:Connexité </a><div class=printfooter>Retrieved from " <a href=http://www.academickids.com/encyclopedia/index.php/Connected_space> http://www.academickids.com/encyclopedia/index.php/Connected_space </a> "</div><div id=catlinks><p class=catlinks><a href=/encyclopedia/index.php?title=Special:Categories&article=Connected_space title=Special:Categories> Categories </a> : <a href=/encyclopedia/index.php/Category:Topology title=Category:Topology> Topology </a> | <a title="Category:General topology"href=/encyclopedia/index.php/Category:General_topology> General topology </a></div><div 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