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# pseudocompact space

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.

Defines:

pseudocompact, pseudocompactness, pseudo-compact, pseudo-compactness, pseudo compact, pseudo compactness

Related:

LimitPointCompact

Synonym:

pseudo compact space, pseudo-compact space

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

54D30*no label found*

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