# Finite Discrete Space satisfies all Compactness Properties

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## 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 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.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1$. Finite Discrete Topology: $8$