Equivalence of Definitions of Locally Path-Connected Space/Definition 3 implies Definition 4

Theorem
Let $T = \struct{S, \tau}$ be a topological space. Let $T$ have a basis consisting of path-connected sets in $T$.

Then
 * the path components of open sets of $T$ are also open in $T$.

Proof
From Path Component of Locally Path-Connected Space is Open, the path components of the open sets of $T$ are also open in $T$.