Countable Open Covers Condition for Separated Sets

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

Let $A, B \subseteq S$

For all $X \subseteq S$, let $X^-$ denote the closure of $X$ in $T$.

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 1:
 * $U \cap V = \O$

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 $T$.