---
tag: FDVAHRJKF3
kind: concept
name: Locally connected space
msc: [54D05]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_connected_space
same_as:
  wikidata: Q865967
  wikipedia: Locally connected space
---

## Summary

A locally connected space is a [[topological space|0CQHFPGJ45]] in which every point has arbitrarily small connected open [[neighbourhoods|0ZCY0Z80B6]].
Equivalently, the [[connected components|0JG9W3KA6B]] of every open subset are open.
Local connectedness and [[connectedness|0V032VWTHD]] are independent: neither implies the other.
It is a hypothesis in many results of continuum theory, such as the [[Hahn–Mazurkiewicz theorem|Q5QFYKD004]].

## Explanations

### L1 human

A space is locally connected if, zooming in near any point, you can always find a small open piece around it that is all in one piece.
Euclidean space $\R^n$ is locally connected, since small open balls are connected.
Two disjoint open intervals form a locally connected space that is not connected.
The [[comb space|A2Q4RXD162]] is connected but not locally connected.