# Connected Space is not necessarily Locally Connected

## Theorem

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

Then it is not necessarily the case that $T$ is also a locally connected space.

## Proof

Let $C$ be the closed topologist's sine curve embedded in the real Euclidean plane.

From Closed Topologist's Sine Curve is Connected, $C$ is connected in $T$

Hence the result.

$\blacksquare$