Discrete Space is Zero Dimensional
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Theorem
Let $T = \struct {S, \tau}$ be a discrete space.
Then $T$ is zero dimensional.
Proof 1
We have from Partition of Singletons yields Discrete Topology that a discrete space is a partition space.
The result follows from Partition Topology is Zero Dimensional.
$\blacksquare$
Proof 2
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
$\blacksquare$
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