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

Theorem
Let $T = \left({S, \tau_{\bar p}}\right)$ 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