Particular Point Space is T0
Theorem
Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is a $T_0$ (Kolmogorov) space.
Proof 1
Let $T$ be a trivial space, so that $S = \set p$.
Then the result holds vacuously as there are no two distinct points in $T$.
Now suppose $T$ is not trivial.
Then $\exists x \in S: x \ne p$.
Now we have that $\set p \subseteq T$ is open in $T$ such that $p \in \set p$ but $x \notin \set p$.
Finally, suppose that $x, y \in S: x \ne y, x \ne p, y \ne p$.
Then we have that $\set {p, x} \subseteq T$ is open in $T$ such that $x \in \set {p, x}$ but $y \notin \set {p, x}$.
Hence the result.
$\blacksquare$
Proof 2
We have:
- Discrete Space satisfies all Separation Properties (including being a $T_0$ space)
Then by Condition for Closed Extension Space to be $T_0$ Space, as a discrete space is $T_0$ then so is its closed extension.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $8 \text { - } 10$. Particular Point Topology: $4$