Discrete Space has Open Locally Finite Cover

Theorem
Let $T = \left({S, \vartheta}\right)$ be a topological space where $\vartheta$ is the discrete topology on $S$.

Then $T$ has an open cover which is locally finite.

Proof
Consider the set:
 * $\mathcal C := \left\{{\left\{{x}\right\}: x \in S}\right\}$

That is, the set of all singleton subsets of $S$.

Then from All Sets in Discrete Topology are Clopen, $\mathcal C$ is an open cover of $T$.

From Points in Discrete Space are Neighborhoods, every point $x \in S$ has a neighborhood $\left\{{x}\right\}$.

This neighborhood $\left\{{x}\right\}$ intersects exactly one element of $\mathcal C$, that is: $\left\{{x}\right\}$ itself.

As $1$ is a finite number, the result follows from definition of locally finite.