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
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: $8$