Countable Open Covers Condition for Separated Sets/Lemma 1

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

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

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

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:
 * ${V_n}' = V_n \setminus \paren {\ds \bigcup_{p \mathop \le n} {U_p}^-}$

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.