Definition:Locally Path-Connected Space

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.