---
tag: HBGCDP2BRN
kind: concept
name: Pseudocompact space
msc: [54D30]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudocompact_space
same_as:
  wikidata: Q5362132
  wikipedia: Pseudocompact space
---

## Summary

A pseudocompact space is a [[topological space|0CQHFPGJ45]] on which every continuous real-valued function is bounded.
It keeps one of the most used consequences of [[compactness|0TVJZYB6SH]], the boundedness of continuous functions, without asking for finite subcovers.
Every compact space and every [[countably compact space|S3N3CHZV89]] is pseudocompact.
For [[metric spaces|0VKD1TP7F3]] the notion gives nothing new: a metric space is pseudocompact exactly when it is compact.

## Explanations

### L1 human

On a closed bounded interval, a continuous function cannot run off to infinity.
A pseudocompact space is a space where this good behaviour holds for every continuous function into the real line.
The real line itself is not pseudocompact, because the function $f(x) = x$ is continuous and unbounded.
The interval $[0, 1]$ is pseudocompact, because it is compact.