Modified Fort Space is not Totally Separated

Theorem

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

Then $T$ is not totally separated.

Proof

We have:

Totally Separated Space is Completely Hausdorff and Urysohn

Completely Hausdorff Space is Hausdorff Space

But we have:

Modified Fort Space is not Hausdorff

The result follows from 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: $5$