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
Let $U$ be an open subset of $T$.

From Open Set is Union of Elements of Basis, $U$ is a union of open path-connected sets in $T$.

From Open Set in Open Subspace and Path-Connected Set in Subspace, $U$ is a union of open path-connected sets in $U$.

From Path Components are Open iff Union of Open Path-Connected Sets, the path components of $U$ are open in $U$.

From Open Set in Open Subspace then the path components of $U$ are open in $T$.