Dispersion Point in Particular Point Space

Theorem

Let $T = \struct {S, \tau_p}$ be a particular point space.


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


Proof

Let $H = S \setminus \set p$.

Let $T_H = \struct {H, \tau_H}$ be the topological subspace induced on $H$ by $\tau_p$.


From Particular Point Space less Particular Point is Discrete, the space $T_H$ is discrete.

We have Discrete Space is Locally Connected.

Thus from Totally Disconnected and Locally Connected Space is Discrete we have that $S \setminus \set p$ is totally disconnected.

Hence the result, 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: $8 \text { - } 10$. Particular Point Topology: $11$
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