Path-Connected Space is not necessarily Locally Path-Connected

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

Then it is not necessarily the case that $T$ is also locally path-connected.

Proof
Let $T$ be the extended topologist's sine curve.

From Extended Topologist's Sine Curve is Path-Connected, $T$ is a path-connected space.

From Extended Topologist's Sine Curve is not Locally Path-Connected, $T$ is not a locally path-connected space.

Hence the result.