Zero Dimensional Space is not necessarily T0

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Theorem

Let $T = \left({S, \tau}\right)$ be a zero dimensional topological space.

Then $T$ is not necessarily a $T_0$ (Kolmogorov) space.


Proof

Let $T = \left({S, \tau}\right)$ be a partition space.

From Partition Topology is Zero Dimensional, $T$ is a zero dimensional topological space.

From Partition Topology is not $T_0$, $T$ is not a $T_0$ (Kolmogorov) space.

$\blacksquare$


Sources