Finite Space Satisfies All Compactness Properties
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Theorem
Let $T = \struct {S, \tau}$ 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:
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
$\blacksquare$