Either-Or Topology is Locally Connected

Theorem

Let $T = \struct {S, \tau}$ be the either-or space.

Then $T$ is a locally connected space.

Proof

Either-Or Topology is Locally Path-Connected

Locally Path-Connected Space is Locally Connected

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $17$. Either-Or Topology: $4$