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Difference between revisions of "Compactness"

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A property which characterizes a wide class of topological spaces, requiring that from any covering of a space by open sets it is possible to extract a finite covering. The term  "bicompactness"  is also used instead of  "compactness"  (in Russian). Topological spaces with the compactness property are called compact spaces (cf. [[Compact space|Compact space]]).
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A property which characterizes a wide class of topological spaces, requiring that from any covering of a space by open sets it is possible to extract a finite covering. Topological spaces with the compactness property are called [[compact space]]s. In Russian literature, "compactness" is often used for the notion of [[Compactness, countable|countable compactness]], and "bicompactness" for general compactness.  Bourbaki uses the term "compact" to include Hausdorff, and more generally uses the term "quasi-compact".
  
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For references, see [[Compact space]].
  
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====References====
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"Compactness" in the Russian literature often refers to the notion of countable compactness.  "Bicompactness" in the Russian literature being the Western notion of  "compactness" .
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> Nicolas Bourbaki, "General Topology: Chapters 1-4", Springer (1998) {{ISBN|3-540-64241-2}}</TD></TR>
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For references, see [[Compact space|Compact space]].
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[[Category:Topology]]

Latest revision as of 17:13, 15 November 2023

2020 Mathematics Subject Classification: Primary: 54D30 [MSN][ZBL]

A property which characterizes a wide class of topological spaces, requiring that from any covering of a space by open sets it is possible to extract a finite covering. Topological spaces with the compactness property are called compact spaces. In Russian literature, "compactness" is often used for the notion of countable compactness, and "bicompactness" for general compactness. Bourbaki uses the term "compact" to include Hausdorff, and more generally uses the term "quasi-compact".

For references, see Compact space.

References

[a1] Nicolas Bourbaki, "General Topology: Chapters 1-4", Springer (1998) ISBN 3-540-64241-2
How to Cite This Entry:
Compactness. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Compactness&oldid=11573
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Compactness&oldid=54471"