---
tag: 1ZVR8BRK65
kind: concept
name: Locally path-connected space
msc: [54D05]
specializes: [FDVAHRJKF3]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_connected_space
---

## Summary

A locally path-connected space is a [[topological space|0CQHFPGJ45]] in which every point has arbitrarily small [[path-connected|0WB6DH499T]] open neighbourhoods.
Every locally path-connected space is [[locally connected|FDVAHRJKF3]].
In a locally path-connected space, the path components and the connected components coincide.
So a connected, locally path-connected space is path-connected, a fact used throughout covering space theory.

## Explanations

### L1 human

Near any point of such a space, a small open piece can be chosen in which every two points are joined by a path inside the piece.
Open subsets of $\R^n$ have this property, because small balls are convex.
The [[comb space|A2Q4RXD162]] fails it: near a point on the left edge high up, nearby points on the teeth cannot be joined by short paths.