Uncountable Discrete Space is not Second-Countable

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

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $3$. Uncountable Discrete Topology: $8$
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