# Finite Discrete Space satisfies all Compactness Properties

## Theorem

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

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 Locally Compact.
$T$ is Weakly Locally Compact.
$T$ is Strongly Locally Compact.
$T$ is $\sigma$-Locally Compact.
$T$ is Weakly $\sigma$-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 fully normal space is fully $T_4$ by definition.

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