# Dispersion Point in Particular Point Space

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## Theorem

Let $T = \left({S, \tau_p}\right)$ be a particular point space.

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

## Proof

Let $H = S \setminus \left\{{p}\right\}$.

Let $T_H = \left({H, \tau_H}\right)$ 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 $S \setminus \left\{{p}\right\}$ is totally disconnected.

Hence the result, from definition of dispersion point.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 8 - 10: \ 11$