Dispersion Point of Excluded Point Space
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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: $5$