Finite Space Satisfies All Compactness Properties

Theorem
Let $T = \left({S, \tau}\right)$ 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.

Proof
We have that:


 * A Finite Topological Space is Compact.
 * A Finite Space is Sequentially Compact.

The remaining properties are demonstrated in:


 * Sequence of Implications of Global Compactness Properties
 * Sequence of Implications of Local Compactness Properties
 * Sequence of Implications of Paracompactness Properties