Countable Discrete Space is Separable

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Theorem

Let $T = \struct {S, \tau}$ be a countable discrete topological space.


Then $T$ is separable.


Proof 1

Let $T = \left({S, \tau}\right)$ be a countable discrete topological space.

From Countable Discrete Space is Second-Countable:

$T$ is second-countable.

From Second-Countable Space is Separable:

$T$ is separable.

$\blacksquare$


Proof 2

Follows immediately from Countable Space is Separable.

$\blacksquare$


Also see


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