Equivalence of Definitions of Locally Path-Connected Space/Definition 1 implies Definition 2

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let each point of $T$ have a local basis consisting entirely of path-connected sets in $T$.

Then

each point of $T$ has a neighborhood basis consisting of path-connected sets in $T$.

Proof

From Local Basis for Open Sets Implies Neighborhood Basis of Open Sets, it follows directly that:

each point of $T$ has a neighborhood basis consisting entirely of path-connected sets in $T$.

$\blacksquare$