Uncountable Excluded Point Space is not Second-Countable

Theorem

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

Then $T$ is not second-countable.

Proof 1

Let $H = S \setminus \left\{{p}\right\}$ where $\setminus$ denotes set difference.

By definition, $H$ is an uncountable discrete space.

The result follows from Uncountable Discrete Space is not Second-Countable.

$\blacksquare$

Proof 2

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

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $15$. Uncountable Excluded Point Topology: $6$