Excluded Point Topology is Open Extension Topology of Discrete Topology

Theorem

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

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

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

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

Proof

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

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $16$. Open Extension Topology: $8$