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The property of a topological space that every infinite subset of it has an accumulation point. For a metric space the notion of countable compactness is the same as that of [[Compactness|compactness]]. The property of countable compactness can be expressed in the following form: Every countable subset has an accumulation point, so that countably compact spaces are naturally called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c0235701.png" />-compact.
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In connection with this arise the concepts of initial and final compactness, or more generally compactness in an interval of cardinals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c0235702.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c0235704.png" />-compactness, expressible in three equivalent forms: 1) each set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c0235705.png" /> of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c0235706.png" /> has a [[Complete accumulation point|complete accumulation point]], that is, a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c0235707.png" /> such that for every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c0235708.png" /> of it, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c0235709.png" /> has the same cardinality as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357010.png" />; 2) every totally ordered system of order type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357011.png" /> of closed sets has a non-empty intersection; and 3) every open covering of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357012.png" /> contains a subcovering of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357013.png" />.
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357014.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357016.png" /> is called initially compact up to cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357018.png" />. Countable compactness means initial compactness up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357019.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357020.png" /> is arbitrary, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357021.png" /> is called finally compact, starting from cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357022.png" />; thus, every space with a countable basis is finally compact from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357023.png" />. Compact spaces are initially compact from any (infinite) cardinality and are at the same time finally compact starting from any cardinality. Thus, every compact space is countably compact, but not conversely: The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357024.png" /> of all ordinal numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357025.png" /> is countably compact, but not compact. The (countable) compactness of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357026.png" /> need not imply that it is sequentially compact. E.g., in the (non-metrizable) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357027.png" /> there exists an infinite closed (and hence Hausdorff compact) set containing no non-stationary convergent subsequence. (Cf. [[Compact set, countably|Compact set, countably]].)
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The property of a topological space that every infinite subset of it has an accumulation point. For a metric space the notion of countable compactness is the same as that of [[Compactness|compactness]]. The property of countable compactness can be expressed in the following form: Every countable subset has an accumulation point, so that countably compact spaces are naturally called  $  \aleph _ {0} $-
 +
compact.
  
 +
In connection with this arise the concepts of initial and final compactness, or more generally compactness in an interval of cardinals  $  [ a , b ] $,
 +
or  $  [ a , b ] $-
 +
compactness, expressible in three equivalent forms: 1) each set  $  M \subset  X $
 +
of cardinality  $  m \in [ a , b ] $
 +
has a [[Complete accumulation point|complete accumulation point]], that is, a point  $  \xi $
 +
such that for every neighbourhood  $  O _  \xi  $
 +
of it, the set  $  O _  \xi  \cap M $
 +
has the same cardinality as  $  M $;
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2) every totally ordered system of order type  $  \omega \in [ \omega _ {a} , \omega _ {b} ] $
 +
of closed sets has a non-empty intersection; and 3) every open covering of cardinality  $  m \in [ a , b ] $
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contains a subcovering of cardinality  $  < m $.
  
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If  $  a $
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equals  $  \aleph _ {0} $,
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then  $  X $
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is called initially compact up to cardinality  $  b $.
 +
Countable compactness means initial compactness up to  $  \aleph _ {0} $.
 +
If  $  b \geq  a $
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is arbitrary, then  $  X $
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is called finally compact, starting from cardinality  $  a $;
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thus, every space with a countable basis is finally compact from  $  a = \aleph _ {1} $.
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Compact spaces are initially compact from any (infinite) cardinality and are at the same time finally compact starting from any cardinality. Thus, every compact space is countably compact, but not conversely: The space  $  W ( \omega _ {1} ) $
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of all ordinal numbers  $  < \omega _ {1} $
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is countably compact, but not compact. The (countable) compactness of a space  $  X $
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need not imply that it is sequentially compact. E.g., in the (non-metrizable) space  $  I ^ { \mathfrak c } $
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there exists an infinite closed (and hence Hausdorff compact) set containing no non-stationary convergent subsequence. (Cf. [[Compact set, countably|Compact set, countably]].)
  
 
====Comments====
 
====Comments====
In particular, the property that each infinite set has an [[Accumulation point|accumulation point]] (or [[Limit point of a set|limit point of a set]], which is the same thing) is equivalent to the property of being countably compact as defined in the article [[Compact space|Compact space]], i.e. in the sense of the property that each countable covering has a finite subcovering. A space is a compact if each infinite subset has a [[Complete accumulation point|complete accumulation point]] (cf. [[#References|[a1]]]). Finally compact (from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357028.png" />) spaces are called Lindelöf spaces in Western terminology.
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In particular, the property that each infinite set has an [[Accumulation point|accumulation point]] (or [[Limit point of a set|limit point of a set]], which is the same thing) is equivalent to the property of being countably compact as defined in the article [[Compact space|Compact space]], i.e. in the sense of the property that each countable covering has a finite subcovering. A space is a compact if each infinite subset has a [[Complete accumulation point|complete accumulation point]] (cf. [[#References|[a1]]]). Finally compact (from $  \aleph _ {1} $)  
 +
spaces are called Lindelöf spaces in Western terminology.
  
In the article above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357029.png" /> is the first ordinal number of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357030.png" />, in standard Western usage, however, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357031.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023570/c02357032.png" />-th infinite cardinal number.
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In the article above $  \omega _ {a} $
 +
is the first ordinal number of cardinality $  a $,  
 +
in standard Western usage, however, $  \omega _ {a} $
 +
is the $  a $-
 +
th infinite cardinal number.
  
 
The Russian terminology on compactness differs from the Western terminology. Compact in Russian literature is countably compact in Western literature, and bicompact equates with compact Hausdorff in the West.
 
The Russian terminology on compactness differs from the Western terminology. Compact in Russian literature is countably compact in Western literature, and bicompact equates with compact Hausdorff in the West.

Revision as of 17:45, 4 June 2020


The property of a topological space that every infinite subset of it has an accumulation point. For a metric space the notion of countable compactness is the same as that of compactness. The property of countable compactness can be expressed in the following form: Every countable subset has an accumulation point, so that countably compact spaces are naturally called $ \aleph _ {0} $- compact.

In connection with this arise the concepts of initial and final compactness, or more generally compactness in an interval of cardinals $ [ a , b ] $, or $ [ a , b ] $- compactness, expressible in three equivalent forms: 1) each set $ M \subset X $ of cardinality $ m \in [ a , b ] $ has a complete accumulation point, that is, a point $ \xi $ such that for every neighbourhood $ O _ \xi $ of it, the set $ O _ \xi \cap M $ has the same cardinality as $ M $; 2) every totally ordered system of order type $ \omega \in [ \omega _ {a} , \omega _ {b} ] $ of closed sets has a non-empty intersection; and 3) every open covering of cardinality $ m \in [ a , b ] $ contains a subcovering of cardinality $ < m $.

If $ a $ equals $ \aleph _ {0} $, then $ X $ is called initially compact up to cardinality $ b $. Countable compactness means initial compactness up to $ \aleph _ {0} $. If $ b \geq a $ is arbitrary, then $ X $ is called finally compact, starting from cardinality $ a $; thus, every space with a countable basis is finally compact from $ a = \aleph _ {1} $. Compact spaces are initially compact from any (infinite) cardinality and are at the same time finally compact starting from any cardinality. Thus, every compact space is countably compact, but not conversely: The space $ W ( \omega _ {1} ) $ of all ordinal numbers $ < \omega _ {1} $ is countably compact, but not compact. The (countable) compactness of a space $ X $ need not imply that it is sequentially compact. E.g., in the (non-metrizable) space $ I ^ { \mathfrak c } $ there exists an infinite closed (and hence Hausdorff compact) set containing no non-stationary convergent subsequence. (Cf. Compact set, countably.)

Comments

In particular, the property that each infinite set has an accumulation point (or limit point of a set, which is the same thing) is equivalent to the property of being countably compact as defined in the article Compact space, i.e. in the sense of the property that each countable covering has a finite subcovering. A space is a compact if each infinite subset has a complete accumulation point (cf. [a1]). Finally compact (from $ \aleph _ {1} $) spaces are called Lindelöf spaces in Western terminology.

In the article above $ \omega _ {a} $ is the first ordinal number of cardinality $ a $, in standard Western usage, however, $ \omega _ {a} $ is the $ a $- th infinite cardinal number.

The Russian terminology on compactness differs from the Western terminology. Compact in Russian literature is countably compact in Western literature, and bicompact equates with compact Hausdorff in the West.

The article above also explains the origin of the Russian term "bicompact topological spacebicompactnessbicompact" . A space is "bicompact" if it is both initially and finally compact.

References

[a1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. Chapt. 3. Problem 48 and pp. 165, 166 (Translated from Russian)
How to Cite This Entry:
Compactness, countable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Compactness,_countable&oldid=46409
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article