---
tag: WQG81SEDZ4
kind: concept
name: Metacompact space
msc: [54D20]
sources:
  - url: https://en.wikipedia.org/wiki/Metacompact_space
same_as:
  wikidata: Q6822437
  wikipedia: Metacompact space
---

## Summary

A metacompact space is a [[topological space|0CQHFPGJ45]] in which every [[open cover|049PXYHW1T]] has a point-finite open refinement.
Point-finite means that each point lies in only finitely many members of the refinement.
This is weaker than [[paracompactness|8R29G6NG9Q]], where each point needs a neighbourhood meeting only finitely many members.
So every paracompact space, and in particular every compact space, is metacompact.

## Explanations

### L1 human

In a metacompact space, you can always shrink an open cover so that no point is covered too many times.
Each single point sits in only finitely many of the new sets.
Points close to it may still be covered by many different sets.
That is the difference from paracompact spaces, where a whole neighbourhood must see only finitely many sets.