---
tag: 1SJ46AX8Y6
kind: definition
name: Locally path-connected space
msc: [54D05]
defines: 1ZVR8BRK65
uses: [0CQHFPGJ45, 02BH3MCBQB, 0WB6DH499T]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_connected_space
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
The space $X$ is *locally path-connected* if the following holds.
For every point $x \in X$ and every [[open set|02BH3MCBQB]] $U$ containing $x$, there is a [[path-connected|0WB6DH499T]] 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|A2Q4RXD162]] 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|FDVAHRJKF3]], because path-connected sets are connected.
The converse fails in general.