Connected Space is not necessarily Locally Connected
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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