Modified Fort Space is not Zero Dimensional

Theorem

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

Then $T$ is not zero dimensional.

Proof

We have Modified Fort Space is not $T_3$.

We also have Zero Dimensional Space is $T_3$.

The result follows by Modus Tollendo Tollens.

$\blacksquare$

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: $5$