Countable Open Covers Condition for Separated Sets

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

Let $A, B \subseteq S$

Let:
 * $\UU = \set{U_n : n \in \N}$ be a countable open cover of $A : \forall n \in \N : U_n^- \cap B = \O$

Let:
 * $\VV = \set{V_n : n \in \N}$ be a countable open cover of $B : \forall n \in \N : V_n^- \cap A = \O$

Then:
 * $A$ and $B$ can be separated in $T$

Proof
By definition of cover:
 * $\ds A \subseteq \bigcup_{n \mathop \in \N} U_n$

We have:
 * $\ds A \cap \paren{\bigcup_{n \mathop \in \N} V_n^-} = \O$

From Subset of Set Difference iff Disjoint Set:
 * $(1) \quad \ds A \subseteq \paren {\bigcup_{n \mathop \in \N} U_n } \setminus \paren {\bigcup_{n \mathop \in \N} V_n^-}$

Similarly:
 * $(2) \quad \ds B \subseteq \paren {\bigcup_{n \mathop \in \N} V_n} \setminus \paren {\bigcup_{n \mathop \in \N} U_n^-}$

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

Lemma 1
Let:
 * $U = \ds \bigcup_{n \mathop \in \N} U'_n$

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

From Lemma 2:
 * $U \cap V = \O$

Lemma 2
We have:

Similarly, from $(2)$ above:

It has been shown:
 * there exists $U, V \in \tau$ such that $A \subseteq U, B \subseteq V$ and $U \cap V = \O$.

Hence, by definition, $A$ and $B$ can be separated in $T$.