T3 Lindelöf Space is T4 Space/Proof 1

Proof
Let $A$ and $B$ be disjoint closed subsets of $T$.

Lemma 1
Let $\UU = \set{U \in \tau : U^- \cap B = \O}$.

From Lemma 1:
 * $\forall a \in A : \exists U_a \in \tau: a \in U_a : U_a \cap B = \O$

By definition of open cover:
 * $\UU$ is an open cover of $A$

From User:Leigh.Samphier/Topology/Closed Subspace of Lindelöf Space is Lindelöf:
 * $\struct{A, \tau_A}$ is a Lindelöf subspace

where $\tau_A$ is the subspace topology on $A$.

By definition of Lindelöf space:
 * there exists a countable subcover $\family{U_{a_n}}_{n \in \N}$ of $\UU$ for $A$

Let $\VV = \set{V \in \tau : V^- \cap A = \O}$.

Similar to $A$ above, there exists a countable subcover $\family{V_n}_{n \in \N}$ of $\VV$ for $B$.

Lemma 1

 * $\ds A \subseteq \paren{\bigcup_{n \in \N} U_{a_n} } \setminus \paren{\bigcup_{n \in \N} V_{a_n}^-}$

Lemma 2

 * $\ds B \subseteq \paren{\bigcup_{n \in \N} V_{a_n}} \setminus \paren{\bigcup_{n \in \N} U_{a_n}^-}$

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

where $V_{a_p}^-$ denote the closure of $V_{a_p}$ in $T$.

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

where $U_{a_p}^-$ denote the closure of $U_{a_p}$ in $T$.

Lemma 3

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

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

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

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

Lemma 4

 * $U$ and $V$ are open in $T$

We have:

and