Fort Space is Compact

Theorem

Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$.

Then $T$ is a compact space.

Proof

Let $\CC$ be an open cover of $T$.

Then $\exists U \in \CC$ such that $p \in U$ and so $\relcomp S U$ is finite.

For each $x \in \relcomp S U$ there exists some $C_x \in \CC$ such that $x \in C$.

So $U$, together with each of those $C_x \in \CC$, is a finite subcover of $\CC$.

Hence the result by definition of compact space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $23 \text { - } 24$. Fort Space: $4$