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