T0 Space is not necessarily T1 Space

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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 1

Proof by Counterexample:

Let $T = \struct {S, \tau_p}$ be a particular point space such that $S$ is not a singleton.


From Particular Point Space is $T_0$, we have that $T$ is a $T_0$ space.

From Non-Trivial Particular Point Topology is not $T_1$, $T$ is not a $T_1$ space.

The result follows.

$\blacksquare$


Proof 2

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.

$\blacksquare$