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