Definition:Locally Path-Connected Space
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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.