Locally Arc-Connected Space is Locally Path-Connected

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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