Discrete Space is Paracompact

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

Then $T$ is paracompact.

Proof
Let $\VV$ be any open cover of $S$.

Consider the set $\CC$ of all singleton subsets of $S$:
 * $\CC := \set {set x: x \in S}$

From Discrete Space has Open Locally Finite Cover, $\CC$ is an open cover which is locally finite.

This result also shows that $\CC$ is the finest cover on $T$.

So $\CC$ is an open refinement of $\VV$ which is locally finite.

So $T$ is paracompact, by definition.