Locally Arc-Connected Space is Locally Path-Connected

Theorem

Let $T = \struct {S, \tau}$ be a topological space which is locally arc-connected.

Then $T$ is also locally path-connected.

Proof

Let $T = \struct {S, \tau}$ be arc-connected.

Then $T$ has a basis consisting entirely of arc-connected sets.

From Arc-Connected Space is Path-Connected, this basis consisting entirely of path-connected sets.

The result follows from definition of locally path-connected.

$\blacksquare$

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