Zero Dimensional Space is not necessarily T0

Theorem

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

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

Proof

Let $T = \struct {S, \tau}$ 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

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness: Disconnectedness