Uncountable Excluded Point Space is not Second-Countable
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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: $6$