Characterization of Paracompactness in T3 Space/Lemma 19

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

Let $\BB$ be a discrete set of subsets of $X$.

Let $\UU = \set{ U \in \tau : \size {\set{B \in \BB : U \cap B} } \le 1}$

Then:
 * $\UU$ is a open cover of $X$ in $T$.

Proof
Let $s \in X$.

By definition of discrete:
 * $\exists U \in \tau : x \in U : \size {\set{B \in \BB : U \cap B} } \le 1$

Hence:
 * $U \in \UU$

Since $x$ was arbitrary:
 * $\forall x \in X : \exists U \in \UU : x \in U$

It follows that $\UU$ is an open cover of $X$ in $T$ by definition.