Countable Discrete Space is Sigma-Compact
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Theorem
Let $T = \struct {S, \tau}$ be a countable discrete topological space.
Then $T$ is $\sigma$-compact.
Proof 1
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.
$\blacksquare$
Proof 2
A direct application of Countable Space is Sigma-Compact.
Also see
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $2$. Countable Discrete Topology: $8$