## You are here

Homepseudocompact space

## Primary tabs

# 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*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff