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Difference between revisions of "Pseudo-compact space"

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A [[Completely-regular space|completely-regular space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075630/p0756301.png" /> such that every real-valued continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075630/p0756302.png" /> is bounded. In the class of normal spaces the concepts of countable compactness (cf. [[Countably-compact space|Countably-compact space]]) and pseudo-compactness coincide.
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A [[Completely-regular space|completely-regular space]] $X$ such that every real-valued continuous function on $X$ is bounded. In the class of normal spaces the concepts of [[Countably-compact space|countable compactness]] and pseudo-compactness coincide.
  
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|valign="top"|{{Ref|ArPo}}||valign="top"| A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises", Reidel (1984) pp. 136 (Translated from Russian)
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 136 (Translated from Russian)</TD></TR></table>
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|valign="top"|{{Ref|En}}||valign="top"| R. Engelking, "General topology", Heldermann (1989)
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Latest revision as of 21:31, 2 May 2012

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

A completely-regular space $X$ such that every real-valued continuous function on $X$ is bounded. In the class of normal spaces the concepts of countable compactness and pseudo-compactness coincide.

References

[ArPo] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises", Reidel (1984) pp. 136 (Translated from Russian)
[En] R. Engelking, "General topology", Heldermann (1989)
How to Cite This Entry:
Pseudo-compact space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-compact_space&oldid=11440
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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