https://wiki.unifiedmathematics.com/index.php?title=A_topological_space_is_disconnected_if_and_only_if_it_is_homeomorphic_to_a_disjoint_union_of_two_or_more_non-empty_topological_spaces&feed=atom&action=historyA topological space is disconnected if and only if it is homeomorphic to a disjoint union of two or more non-empty topological spaces - Revision history2024-03-28T22:15:41ZRevision history for this page on the wikiMediaWiki 1.24.1https://wiki.unifiedmathematics.com/index.php?title=A_topological_space_is_disconnected_if_and_only_if_it_is_homeomorphic_to_a_disjoint_union_of_two_or_more_non-empty_topological_spaces&diff=3131&oldid=prevAlec: Created page with "__TOC__ ==Statement== Let {{Top.|X|J}} be a topological space, then{{rITTMJML}}: * {{Top.|X|J}} is {{link|disconnected|topology}} (ie: not {{link|connected|topology}}) {{i..."2016-09-30T23:15:13Z<p>Created page with "__TOC__ ==Statement== Let {{Top.|X|J}} be a <a href="/index.php?title=Topological_space" title="Topological space">topological space</a>, then{{rITTMJML}}: * {{Top.|X|J}} is {{link|disconnected|topology}} (ie: not {{link|connected|topology}}) {{i..."</p>
<p><b>New page</b></p><div>__TOC__<br />
==Statement==<br />
Let {{Top.|X|J}} be a [[topological space]], then{{rITTMJML}}:<br />
* {{Top.|X|J}} is {{link|disconnected|topology}} (ie: not {{link|connected|topology}}) {{iff}} it is [[homeomorphic]] to a [[disjoint union topology|disjoint union]] of two or more non-empty topological spaces<br />
==Proof==<br />
{{Requires proof|grade=C|easy=true|msg=Exercise in Lee's top. manifolds, didn't take me very long to do, note that [[A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements]] is a useful "precursor" theorem}}<br />
==See also==<br />
* [[A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements]]<br />
==References==<br />
<references/><br />
{{Theorem Of|Topology}}</div>Alec