Path-Connected Space is not necessarily Locally Path-Connected

Theorem

Let $T = \struct {S, \tau}$ 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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness