Excluded Point Topology is Open Extension Topology of Discrete Topology

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Theorem

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

Let $\tau_{\bar p}$ be the excluded point topology on $S$.


Let $T = \left({S \setminus \left\{{p}\right\}, \vartheta}\right)$ be the discrete topological space on $S \setminus \left\{{p}\right\}$.


Then $T^* = \left({S, \tau_{\bar p}}\right)$ is an open extension space of $T$.


Proof

Directly apparent from the definitions of excluded point topology, discrete topological space and open extension space.

$\blacksquare$


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