Locally connected space

Builds on Topological space Open set Connected set

Definition

Let $X$ be a topological space. The space $X$ is locally connected if the following holds. For every point $x \in X$ and every open set $U$ containing $x$ , there is a connected 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 of every open subset of $X$ is open in $X$ .

Reading path

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Connected set concept
Open set concept
Topological space concept
Locally connected space definition

Tag	`D4MRA2R7N8`
URI	`m2://D4MRA2R7N8`
Kind	definition
Writing	W0 stub
MSC	54D05
Defines	Locally connected space
Uses	Topological space, Open set, Connected set
Changed	2 November 2026

Locally connected space

Definition

Remarks

Reading path

Further reading

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What links here