Finite Space Satisfies All Compactness Properties

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