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 $T$

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

Hence the result.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness
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