# Particular Point Topology is Closed Extension Topology of Discrete Topology

## Theorem

Let $S$ be a set and let $p \in S$.

Let $\tau_p$ be the particular point topology on $S$.

Let $T = \struct {S \setminus \set p, \vartheta}$ be the discrete topological space on $S \setminus \set p$.

Then $T^* = \struct {S, \tau_p}$ is a closed extension space of $T$.

## Proof

Directly apparent from the definitions of particular point topology, discrete topological space and closed extension space.

$\blacksquare$