Uncountable Discrete Space is not Second-Countable

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Theorem

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


Then $T$ is not second-countable.


Proof

We have that an Uncountable Discrete Space is not Separable.

From Second-Countable Space is Separable, it follows that $T$ can not be second-countable.

$\blacksquare$


Also see


Sources