T0 Space is not necessarily T1 Space/Proof 2

Theorem

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

Then it is not necessarily the case that $T$ is a $T_1$ space.

Proof

Proof by Counterexample:

Let $T$ be the overlapping interval space.

From Existence of Topological Space which satisfies no Separation Axioms but $T_0$, we have that:

$T$ is a $T_0$ space

but:

$T$ is not a $T_1$ space.

The result follows.

$\blacksquare$