Finite Space is Sequentially Compact
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Theorem
Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set.
Then $T$ is sequentially compact.
Proof
We have:
- Finite Topological Space is Compact
- Compact Space is Countably Compact
- Finite Space is Second-Countable
- Second-Countable Space is First-Countable
- First-Countable Space is Sequentially Compact iff Countably Compact.
Hence the result.
$\blacksquare$