# Zero Dimensional T0 Space is Totally Separated

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## Theorem

Let $T = \struct {S, \tau}$ be a zero dimensional topological space which is also a $T_0$ (Kolmogorov) space.

Then $T$ is totally separated.

## Proof

Let $T = \struct {S, \tau}$ be a zero dimensional space which is also a $T_0$ (Kolmogorov) space.

As $T$ is zero dimensional, there exists a basis $\BB$ whose sets are all clopen.

Let $x, y \in S$.

As $T$ is a $T_0$ space:

- $\exists U \in \tau: x \in U, y \notin U$

or:

- $\exists U \in \tau: y \in U, x \notin U$

Without loss of generality, suppose that $\exists U \in \tau: x \in U, y \notin U$.

Then:

- $\exists V \in \BB: x \in V$ and $V \subseteq U$

by definition of basis.

The set $V$ is clopen by the definition of $\BB$.

But then $x \in V$ which is open and $y \in S \setminus V$ which is also open.

$\set {V \mid S \setminus V}$ is a partition and hence $T$ is totally separated.

$\blacksquare$

## 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