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pseudocompact space (Definition)

A topological space $ X$ is said to be pseudocompact if every continuous function $ f\colon X\to\mathbb{R}$ has bounded image.

All countably compact spaces (which includes all compact spaces and all sequentially compact spaces) are pseudocompact. A metric space is pseudocompact if and only if it is compact. A Hausdorff normal space is pseudocompact if and only if it is countably compact.



"pseudocompact space" is owned by yark.
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See Also: weakly countably compact

Other names:  pseudo compact space, pseudo-compact space
Also defines:  pseudocompact, pseudocompactness, pseudo-compact, pseudo-compactness, pseudo compact, pseudo compactness
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Cross-references: normal space, Hausdorff, metric space, sequentially compact, compact, countably compact, image, bounded, continuous function, topological space
There are 4 references to this entry.

This is version 4 of pseudocompact space, born on 2004-04-29, modified 2007-05-23.
Object id is 5815, canonical name is PseudocompactSpace.
Accessed 5758 times total.

Classification:
AMS MSC: 54D30 (General topology :: Fairly general properties :: Compactness)
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