Condition for Open Extension Space to be T0 Space

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.

Then $T^*_{\bar p}$ is a $T_0$ (Kolmogorov) space $T$ is.

Proof
By definition:


 * $\tau^*_{\bar p} = \set {U: U \in \tau} \cup \set {S^*_p}$

Let $T = \struct {S, \tau}$ be a $T_0$ space.

Let $\forall x, y \in S$ such that $x \ne y$.

First, suppose $x \ne p \ne y$.

Then by definition of $T$ as a $T_0$ space:
 * $\exists U \in \tau: x \in U, y \notin U$

or:
 * $\exists U \in \tau: y \in U, x \notin U$

Then as $U \in \tau^*_{\bar p}$, it follows that:


 * $\exists U \in \tau^*_{\bar p}: x \in U, y \notin U$

or:
 * $\exists U \in \tau^*_{\bar p}: y \in U, x \notin U$

Now suppose that $y = p$.

Then by definition:
 * $\forall U \in \tau^*_{\bar p}: x \in U, p \notin U$

unless $U = S^*_p$.

So we have shown that:
 * $\forall x, y \in S^*_p$ such that $x \ne y$, either:
 * $\exists U \in \tau^*_{\bar p}: x \in U, y \notin U$
 * or:
 * $\exists U \in \tau^*_{\bar p}: y \in U, x \notin U$

and so $T^*_{\bar p}$ is a $T_0$ space.

Now suppose that $T = \struct {S, \tau}$ is not a $T_0$ space.

Let $x, y \in S$ such that:


 * $\forall U \in \tau: x \in U \implies y \in U$
 * $\forall U \in \tau: y \in U \implies x \in U$

As $U \in \tau^*_{\bar p}$ it follows that:
 * $\forall U \in \tau^*_{\bar p}: x \in U \implies y \in U$
 * $\forall U \in \tau^*_{\bar p}: y \in U \implies x \in U$

and if $U = S^*_p$ the same applies.

Hence $T^*_{\bar p}$ is not a $T_0$ space.