Dispersion Point of Excluded Point Space

Theorem

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


Then $p$ is a dispersion point of $T$.


Proof

We have that the Excluded Point Topology is Open Extension Topology of Discrete Topology.

So $S \setminus \set p$ is a discrete space.

Then a discrete space is totally disconnected.

The result follows from definition of dispersion point.

$\blacksquare$


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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