Arc-Connected Space is not necessarily Locally Arc-Connected

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Theorem

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


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


Proof

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

From Extended Topologist's Sine Curve is Arc-Connected, $T$ is an arc-connected space.

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

Hence the result.

$\blacksquare$


Sources