https://topospaces.subwiki.org/w/index.php?action=history&feed=atom&title=US_not_implies_HausdorffUS not implies Hausdorff - Revision history2026-04-21T00:40:21ZRevision history for this page on the wikiMediaWiki 1.41.2https://topospaces.subwiki.org/w/index.php?title=US_not_implies_Hausdorff&diff=2947&oldid=prevVipul: Created page with '{{topospace property non-implication| stronger = US-space| weaker = Hausdorff space}} ==Statement== It is possible to have a US-space (i.e., a topological space in which ev…'2009-10-26T18:49:58Z<p>Created page with '{{topospace property non-implication| stronger = US-space| weaker = Hausdorff space}} ==Statement== It is possible to have a <a href="/wiki/US-space" title="US-space">US-space</a> (i.e., a topological space in which ev…'</p>
<p><b>New page</b></p><div>{{topospace property non-implication|<br />
stronger = US-space|<br />
weaker = Hausdorff space}}<br />
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==Statement==<br />
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It is possible to have a [[US-space]] (i.e., a topological space in which every convergent sequence has at most one [[limit]]) that is not a [[Hausdorff space]].<br />
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==Related facts==<br />
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* [[T1 not implies Hausdorff]]<br />
* [[T1 not implies US]]<br />
* [[US implies T1]]<br />
* [[Hausdorff implies US]]<br />
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==Proof==<br />
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===Example of cofinite topology===<br />
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Consider a countable set, say <math>\{ 1,2,3,\dots \}</math>, equipped with the [[cofinite topology]]. With this topology, the set is a US-space, because by definition, the only convergent sequence are those that are eventually constant, with the unique limit being the eventual constant value. However, the space is not Hausdorff, because for any two distinct points <math>x,y</math>, and open sets containing <math>x</math> and <math>y</math>, the open sets intersect. (another way of thinking of this is that the space is an [[irreducible space]]).</div>Vipul