# Definition:Locally Connected Space

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## Contents

## Definition

### Definition 1: Using Local Bases

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

### Definition 2: Using Neighborhood Bases

A topological space $T = \struct{S, \tau}$ is **locally connected** if and only if $T$ is weakly locally connected at each point of $T$.

### Definition 3: Using (Global) Basis

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

### Definition 4: Using Open Components

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

## Also see

- Equivalence of Definitions of Locally Connected Space
- Definition:Weakly Locally Connected at Point
- Definition:Locally Path-Connected Space

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