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.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $12$. Closed Extension Topology: $20$