Countable Discrete Space is Sigma-Compact/Proof 1
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Theorem
Let $T = \struct {S, \tau}$ be a countable discrete topological space.
Then $T$ is $\sigma$-compact.
Proof
We have that Singleton Set in Discrete Space is Compact.
We also have that $S$ is the union of all its singleton sets:
- $\ds S = \bigcup_{x \mathop \in S} \set x$
As $S$ is countable, it is the union of countably many compact sets.
Hence the result, by definition of $\sigma$-compact.
$\blacksquare$