Pseudocompact space

Builds on Topological space Continuous map

Definition

Let $X$ be a topological space. The space $X$ is pseudocompact if every continuous map $f \colon X \to \R$ is bounded, that is, there is a real number $M$ with $\abs{f(x)} \le M$ for all $x \in X$ .

Remarks

Example: a compact space $X$ is pseudocompact. The image $f(X)$ of a continuous $f \colon X \to \R$ is compact in $\R$ , so it is bounded.

Non-example: the real line $\R$ is not pseudocompact. The identity map $f(x) = x$ is continuous and takes arbitrarily large values.

Some authors require in addition that $X$ be a Tychonoff space. A normal Hausdorff pseudocompact space is countably compact.

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Tag	`WM4QWH5F65`
URI	`m2://WM4QWH5F65`
Kind	definition
Writing	W0 stub
MSC	54D30
Defines	Pseudocompact space
Uses	Topological space, Continuous map
Changed	2 November 2026

Pseudocompact space

Definition

Remarks

Reading path

Further reading

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