---
tag: D4MRA2R7N8
kind: definition
name: Locally connected space
msc: [54D05]
defines: FDVAHRJKF3
uses: [0CQHFPGJ45, 02BH3MCBQB, 0SPE8WNWFV]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_connected_space
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *locally connected* if the following holds.
For every point $x \in X$ and every [[open set|02BH3MCBQB]] $U$ containing $x$, there is a [[connected|0SPE8WNWFV]] open set $V$ with $x \in V \subseteq U$.

## Remarks

Example: $\R$ is locally connected.
For $x \in U$ with $U$ open, some interval $(x - \varepsilon, x + \varepsilon)$ lies in $U$, and intervals are connected.

Non-example: $\Q$ with the subspace topology from $\R$ is not locally connected.
Its only nonempty connected subsets are single points, and single points are not open in $\Q$.

Equivalently, $X$ is locally connected if and only if every [[connected component|0JG9W3KA6B]] of every open subset of $X$ is open in $X$.