Connected Space is not necessarily Locally Connected

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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.

$\blacksquare$


Sources