---
tag: 8R29G6NG9Q
kind: concept
name: Paracompact space
msc: [54D20]
specializes: [WQG81SEDZ4]
sources:
  - url: https://en.wikipedia.org/wiki/Paracompact_space
same_as:
  wikidata: Q970119
  wikipedia: Paracompact space
---

## Summary

A paracompact space is a [[topological space|0CQHFPGJ45]] in which every [[open cover|049PXYHW1T]] has an open refinement that is a [[locally finite family|DZ297MMGV8]].
Every [[compact space|0TVJZYB6SH]] and every [[metric space|0VKD1TP7F3]] is paracompact.
Paracompact Hausdorff spaces are [[normal|0NSR9V8TED]], and they admit partitions of unity subordinate to any open cover.
This is why paracompactness is a standard assumption on manifolds: it lets one glue local constructions into global ones.

## Explanations

### L1 human

Compactness asks that every open cover can be cut down to finitely many sets.
Paracompactness asks less: every open cover can be replaced by smaller open sets that are only "finitely many near each point".
On the real line, any open cover can be refined to small intervals that pile up only finitely often around each point.
This local finiteness is enough to add up infinitely many local pieces without trouble.