Discrete Space is Zero Dimensional/Proof 2

Theorem

Let $T = \struct {S, \tau}$ be a discrete space.

Then $T$ is zero dimensional.

Proof

Let $\BB$ be the set:

$\BB := \set {\set x: x \in S}$

From Basis for Discrete Topology, $\BB$ is a basis for $T$.

From Set in Discrete Topology is Clopen, all the elements of $\BB$ are both closed and open.

Hence the result, by definition of zero dimensional space

$\blacksquare$