Dispersion Point in Particular Point Space
Jump to navigation
Jump to search
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: $11$