Characterization of Paracompactness in T3 Space/Lemma 5

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Theorem

Let $X$ be a set.

Let $\AA$ and $\VV$ be sets of subsets of $X$.

For each $V \in \VV$, let:

$V^* = X \setminus \ds \bigcup \set{A \in \AA | A \cap V = \O}$

Then:

$\forall V \in \VV: V \subseteq V^*$

Proof

Let $V \in \VV$.

Let $\AA_V = \set{A \in \AA | A \cap V = \O}$.

From Subset of Set Difference iff Disjoint Set:

$\forall A \in \AA_V : V \subseteq X \setminus A$

We have:

\(\ds V\)

\(\subseteq\)

\(\ds \bigcap \set{X \setminus A : A \in \AA_V}\)

Set is Subset of Intersection of Supersets

\(\ds \)

\(=\)

\(\ds X \setminus \bigcup \set{A : A \in \AA_V}\)

De Morgan's Laws for Set Difference

\(\ds \)

\(=\)

\(\ds V^*\)

definition of $V^*$

$\blacksquare$