Open Subset of Locally Path-Connected Space is Locally Path-Connected

Theorem
Let $T = \left({S, \tau}\right)$ be a locally path-connected topological space.

Let $U \subset S$ be open in $T$.

Then $U$ is locally path-connected in $T$.

Also see

 * Open Subset of Locally Connected Space is Locally Connected, an analogous result for connected components