Uncountable Excluded Point Space is not Second-Countable/Proof 2

Theorem

Let $T = \struct {S, \tau_{\bar p} }$ be an uncountable excluded point space.

Then $T$ is not second-countable.

Proof

We have:

Uncountable Discrete Space is not Second-Countable

Excluded Point Topology is Open Extension Topology of Discrete Topology

The result follows from Condition for Open Extension Space to be Second-Countable

$\blacksquare$