Characterization of Paracompactness in T3 Space/Lemma 19

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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.

$\blacksquare$