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In mathematics, and especially [[topology]], a '''Pytkeev space''' is a [[topological space]] that satisfies qualities more subtle than a [[convergence (mathematics)|convergence]] of a [[sequence]].

== Definitions ==

Let ''X'' be a topological space. For a subset ''S'' of ''X'' let ''S'' denote the [[closure (topology)|closure]] of ''S''. Then a point ''x'' is called a ''Pytkeev point'' if for every set A with {{nowrap|1=''x'' ∈ ''A'' \ {''x''}}}, there is a countable <math> \pi </math>-net of infinite subsets of ''A''. A ''Pytkeev space'' is a space in which every point is a Pytkeev point.<ref name="TAIA">{{cite journal|last=Malykhin|first=V. I.|last2=Tironi|first2=G|year=2000|title=Weakly Fréchet–Urysohn and Pytkeev spaces|journal=Topology and its Applications|volume=104|issue=2|pages=181 − 190}}</ref>

==Examples==

* Every [[sequential space]] is also a Pytkeev space. This is because, if {{nowrap|1=''x'' ∈ ''A'' \ {''x''}}} then there exists a sequence {''ak''} that converges to ''x''. So take the countable π-net of infinite subsets of ''A'' to be {{nowrap|1={''Ak''} = {''ak'', ''ak''+1, ''ak''+2, …}}}.<ref name="TAIA"/>

* If ''X'' is a Pytkeev space, then it is also a [[Weakly Fréchet–Urysohn space]].

== References ==

{{Reflist}}

[[Category:Topology]]

{{topology-stub}}

Periodic systems of small molecules - Revision history

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