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
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
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.
$\blacksquare$