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Difference between revisions of "Compact set, countably"

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A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023520/c0235201.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023520/c0235202.png" /> that as a subspace of this space is countably compact (cf. [[Compact space, countably|Compact space, countably]]). Countable compactness means that every sequence has an [[Accumulation point|accumulation point]], i.e. a point every neighbourhood of which contains infinitely many terms of the sequence.
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A set $M$ in a topological space $X$ that as a subspace of this space is countably compact (cf. [[Compact space, countably|Compact space, countably]]). Countable compactness means that every sequence has an [[Accumulation point|accumulation point]], i.e. a point every neighbourhood of which contains infinitely many terms of the sequence.
  
A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023520/c0235203.png" /> is called sequentially compact if every sequence has a converging subsequence, i.e. if every sequence has a subsequence converging to some point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023520/c0235204.png" />.
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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 $X$.
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023520/c0235205.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023520/c0235206.png" /> is called relatively (sequentially, countably) compact if its closure has the corresponding property.
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A set $M$ in a topological space $X$ is called relatively (sequentially, countably) compact if its closure has the corresponding property.
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023520/c0235207.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023520/c0235208.png" /> such that every infinite sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023520/c0235209.png" /> has a subsequence converging to some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023520/c02352010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023520/c02352011.png" /> (respectively, has an accumulation point) could be called conditionally sequentially compact (respectively, conditionally countably compact).
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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).
  
 
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====References====
 
====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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<table>
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<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>
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Revision as of 20:40, 2 November 2014

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 $X$.

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 spaces and Banach spaces with the weak topology the notions of compactness, sequential compactness and countable compactness coincide.

References

[a1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
How to Cite This Entry:
Compact set, countably. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Compact_set,_countably&oldid=34250