Particular Point Topology is Closed Extension Topology of Discrete Topology

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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.

$\blacksquare$


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