Discrete Space is Paracompact

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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.

$\blacksquare$


Sources