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