Excluded Point Space is Scattered

Theorem

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


Then $T$ is a scattered space.


Proof

We have that Subset of Excluded Point Space is not Dense-in-itself.

So, by definition, $T$ is scattered.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $13 \text { - } 15$. Excluded Point Topology: $5$
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