Countable Open Covers Condition for Separated Sets/Lemma 2

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

Let $\family {U_n}_{n \mathop \in \N}$ be a countable family of open sets.

Let $\family {V_n}_{n \mathop \in \N}$ be a countable family of open sets.

For each $n \in \N$, let:
 * ${U_n}' = U_n \setminus \paren {\ds \bigcup_{p \mathop \le n} {V_p}^-}$

For each $n \in \N$, let:
 * ${V_n}' = V_n \setminus \paren {\ds \bigcup_{p \mathop \le n} {U_p}^-}$

Let:
 * $U = \ds \bigcup_{n \mathop \in \N} {U_n}'$

and
 * $V = \ds \bigcup_{n \mathop \in \N} {V_n}'$

Then:
 * $U$ and $V$ are open in $T$.

Proof
By, it is sufficient to show that:
 * $\forall n \in \N : {U_n}', {V_n}' \in \tau$

Let $n \in \N$.

We have:

From Topological Closure is Closed:
 * ${V_p}^-$ is a closed set.

By definition of closed set:
 * $\relcomp S {V_p^-}$ is an open set.

Hence:
 * ${U_n}'$ is the finite intersection of open sets.

By :
 * ${U_n}'$ is an open set.

Similarly:
 * ${V_n}'$ is an open set.

The result follows from.