Zero Dimensional T0 Space is Totally Separated

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Theorem

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

Then $T$ is totally separated.


Proof

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

As $T$ is zero dimensional, there exists a basis $\mathcal B$ 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:

$\displaystyle \exists V \in \mathcal B: x \in V$ and $V \subseteq U$

by definition of basis.

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

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

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

$\blacksquare$


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