Excluded Point Space is Compact

Theorem

Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space.

Then $T$ is a compact space.

Proof 1

We have:

Excluded Point Topology is Open Extension Topology of Discrete Topology

Open Extension Space is Compact

$\blacksquare$

Proof 2

By definition of excluded point space, the only open set of $T$ which contains $p$ is $S$.

So any open cover $\CC$ of $T$ must have $S$ in it.

So $\set S$ will be a subcover of $\CC$, whatever $\CC$ may be.

And $\set S$ (having only one set in it) is trivially a finite cover of $T$.

Hence the result, by definition of compact topological space.

$\blacksquare$