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Difference between revisions of "Strongly countably complete topological space"

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A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306001.png" /> for which there is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306002.png" /> of open coverings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306003.png" /> such that a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306004.png" /> of closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306005.png" /> has a non-empty intersection whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306006.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306007.png" /> and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306008.png" /> is a subset of some member of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s1306009.png" />.
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A topological space $X$ for which there is a sequence $\{\mathcal A_i\}$ of open coverings of $X$ such that a sequence $\{F_i\}$ of closed subsets of $X$ has a non-empty intersection whenever $F_i\supset F_{i+1}$ for all $i$ and each $F_i$ is a subset of some member of $\mathcal A_i$.
  
 
Locally countably compact spaces and Čech-complete spaces are strongly countably complete. Every strongly countably complete space is a [[Baire space|Baire space]] (but not vice versa).
 
Locally countably compact spaces and Čech-complete spaces are strongly countably complete. Every strongly countably complete space is a [[Baire space|Baire space]] (but not vice versa).
  
This rather technical notion plays an important role in questions whether separate continuity of a mapping on a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s13060010.png" /> implies joint continuity on a large subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130600/s13060011.png" />, see [[Namioka space|Namioka space]]; [[Namioka theorem|Namioka theorem]]; [[Separate and joint continuity|Separate and joint continuity]]; or [[#References|[a2]]].
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This rather technical notion plays an important role in questions whether separate continuity of a mapping on a product $X\times Y$ implies joint continuity on a large subset of $X\times Y$, see [[Namioka space|Namioka space]]; [[Namioka theorem|Namioka theorem]]; [[Separate and joint continuity|Separate and joint continuity]]; or [[#References|[a2]]].
  
 
Strongly countably complete topological spaces were introduced by Z. Frolík, [[#References|[a1]]].
 
Strongly countably complete topological spaces were introduced by Z. Frolík, [[#References|[a1]]].

Latest revision as of 20:11, 12 July 2014

A topological space $X$ for which there is a sequence $\{\mathcal A_i\}$ of open coverings of $X$ such that a sequence $\{F_i\}$ of closed subsets of $X$ has a non-empty intersection whenever $F_i\supset F_{i+1}$ for all $i$ and each $F_i$ is a subset of some member of $\mathcal A_i$.

Locally countably compact spaces and Čech-complete spaces are strongly countably complete. Every strongly countably complete space is a Baire space (but not vice versa).

This rather technical notion plays an important role in questions whether separate continuity of a mapping on a product $X\times Y$ implies joint continuity on a large subset of $X\times Y$, see Namioka space; Namioka theorem; Separate and joint continuity; or [a2].

Strongly countably complete topological spaces were introduced by Z. Frolík, [a1].

References

[a1] Z. Frolík, "Baire spaces and some generalizations of complete metric spaces" Czech. Math. J. , 11 (1961) pp. 237–248
[a2] I. Namioka, "Separate continuity and joint continuity" Pacific J. Math. , 51 (1974) pp. 515–531
How to Cite This Entry:
Strongly countably complete topological space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strongly_countably_complete_topological_space&oldid=32427
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article