Locally path-connected space

Builds on Topological space Open set Path-connected space

Definition

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

Remarks

Example: every open subset $W$ of $\R^n$ is locally path-connected. For $x \in U \subseteq W$ with $U$ open, some open ball around $x$ lies in $U$ , and a ball is path-connected by straight segments.

Non-example: the comb space is not locally path-connected. Small open sets around $(0, 1)$ contain points of the teeth $\set{1/n} \times [0, 1]$ , and these cannot be joined to $(0, 1)$ by a path inside the small set.

Every locally path-connected space is locally connected, because path-connected sets are connected. The converse fails in general.

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Tag	`1SJ46AX8Y6`
URI	`m2://1SJ46AX8Y6`
Kind	definition
Writing	W0 stub
MSC	54D05
Defines	Locally path-connected space
Uses	Topological space, Open set, Path-connected space
Changed	2 November 2026

Locally path-connected space

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here