View source for Compact set, countably

A set $M$ in a topological space $X$ that as a subspace of this space is countably compact (cf. [[Compact space, countably]]). Countable compactness means that every sequence has an [[accumulation point]], i.e. a point every neighbourhood of which contains infinitely many terms of the sequence.

A topological space $X$ is called sequentially compact if every sequence has a converging subsequence, i.e. if every sequence has a subsequence converging to some point of $X$ (cf. [[Sequentially-compact space]]).

A set $M$ in a topological space $X$ is called relatively (sequentially, countably) compact if its closure has the corresponding property.

A set $M$ in a topological space $X$ such that every infinite sequence $\{ x_i : i \in \mathbb{Z}\,,\ x_i \in M \}$ has a subsequence converging to some point $x_0$ of $X$ (respectively, has an accumulation point) could be called conditionally sequentially compact (respectively, conditionally countably compact).

====Comments====
In [[metric space]]s and [[Banach space]]s with the [[weak topology]] the notions of compactness, sequential compactness and countable compactness coincide.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR>
</table>

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[[Category:General topology]]

{{SourceattributionEOM|Compact set, countably}}
[[Category:Mathematics]]