Dispersion Point in Particular Point Space

From ProofWiki
Jump to navigation Jump to search

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