# Definition:Locally Path-Connected Space

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## Contents

## Definition

### Definition 1: using Local Bases

A topological space $T = \struct{S, \tau}$ is a **locally path-connected space** if and only if each point of $T$ has a local basis consisting of path-connected sets in $T$.

### Definition 2: using Neighborhood Bases

A topological space $T = \struct{S, \tau}$ is a **locally path-connected space** if and only if each point of $T$ has a neighborhood basis consisting of path-connected sets in $T$.

### Definition 3: using (Global) Basis

A topological space $T = \struct{S, \tau}$ is a **locally path-connected space** if and only if it has a basis consisting of path-connected sets in $T$.

### Definition 4: using Open Path Components

A topological space $T = \struct{S, \tau}$ is a **locally path-connected space** if and only if the path components of open sets of $T$ are also open in $T$.

## Also see

- Results about
**locally path-connected spaces**can be found here.