Modified Fort Space is not Locally Connected

Theorem

Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.

Then $T$ is not locally connected.

Proof

We have:

Modified Fort Space is Totally Disconnected

Totally Disconnected and Locally Connected Space is Discrete

We also have:

Modified Fort Space is not $T_2$

Discrete Space satisfies all Separation Properties

Hence modified Fort space is not the discrete space, and the result follows.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $27$. Modified Fort Space: $4$