Countable Open Covers Condition for Separated Sets/Lemma 1

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

Let $\family{U_n}_{n \in \N}$ be a family of subsets of $S$.

Let $\family{V_n}_{n \in \N}$ be a family of subsets of $S$.

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

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

Then:
 * $\forall n, m \in \N : U'_n \cap V'_m = \O$

Proof
Let $n, m \in \N$.

, let $m \le n$.

We have:

As $m \le n$, then:
 * $U_m^- \in \set{U_p^- : p \le n}$

We have:

Also:

Finally:

From Empty Set is Subset of All Sets:
 * $\O \subseteq U'_m \cap V'_n $

By definition of set equality:
 * $U'_m \cap V'_n = \O$

Similarly:
 * $U'_n \cap V'_m = \O$

The result follows.