Finite Space Satisfies All Compactness Properties

Theorem

Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set.

Then $T$ satisfies the following compactness properties:

$T$ is compact.

$T$ is sequentially compact.

$T$ is countably compact.

$T$ is weakly countably compact.

$T$ is a Lindelöf space.

$T$ is pseudocompact.

$T$ is $\sigma$-compact.

$T$ is strongly locally compact.

$T$ is $\sigma$-locally compact.

$T$ is weakly $\sigma$-locally compact.

$T$ is locally compact.

$T$ is weakly locally compact.

$T$ is paracompact.

$T$ is countably paracompact.

$T$ is metacompact.

$T$ is countably metacompact.

We have that:

The remaining properties are demonstrated in:

$\blacksquare$