Finite Discrete Space satisfies all Compactness Properties

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the discrete topology on $S$.

Let $S$ be an finite set, thereby making $\vartheta$ the finite discrete topology on $S$.

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 Lindelöf Space
 * $T$ is Pseudocompact.
 * $T$ is $\sigma$-Compact.
 * $T$ is Strongly Locally Compact.
 * $T$ is $\sigma$-Locally Compact.
 * $T$ is Locally Compact.
 * $T$ is Fully Normal.
 * $T$ is Fully $T_4$.
 * $T$ is Paracompact.
 * $T$ is Countably Paracompact.
 * $T$ is Metacompact.
 * $T$ is Countably Metacompact.

Proof

 * A finite discrete space is by definition a topology on a finite set.


 * A Discrete Space is Fully Normal.
 * A fully normal space is fully $T_4$ by definition.

The rest of the results follow directly from Finite Space Satisfies All Compactness Properties.