Definition:Locally Connected Space

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.