Connected Space is not necessarily Locally Connected

Theorem
Let $T = \left({S, \tau}\right)$ 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 $T$

From Closed Topologist's Sine Curve is not Locally Connected, $C$ is not locally connected.

Hence the result.