Equivalence of Definitions of T4 Space

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

The following two conditions defining a $T_4$ space are logically equivalent:

Definition by Open Sets
$T$ is a $T_4$ space iff:


 * $\forall A, B \in \complement \left({\vartheta}\right), A \cap B = \varnothing: \exists U, V \in \vartheta: A \subseteq U, B \subseteq V, U \cap V = \varnothing$

That is, for any two disjoint closed sets $A, B \subseteq X$ there exist disjoint open sets $U, V \in \vartheta$ containing $A$ and $B$ respectively.

Definition by Closed Neighborhoods
$T$ is a $T_4$ space iff each open set $U$ contains a closed neighborhood of each closed set contained in $U$.

Definition by Open Sets implies Definition by Closed Neighborhoods
Let $T$ satisfy the definition by open sets of a $T_4$ space.

Let $A$ be a closed set in $T$, and let $U_A$ be an open neighborhood of $A$.

Then $A$ and $B := \complement \left({U_A}\right)$ are disjoint closed sets in $T$, by Intersection of Complement with Subset is Empty.

Also, from Complements Invert Subsets, $B \subseteq \complement \left({A}\right)$.

By assumption, there exist disjoint open sets $U$ and $V$ such that $A \subseteq U$ and $B \subseteq V$.

From Intersection Preserves Subsets, $B \subseteq \complement \left({A}\right) \cap V$.

Note that the latter set, being an intersection of open sets, is itself open.

Then, from Complement of Complement, De Morgan's Laws: Complement and Complements Invert Subsets:


 * $A \cup \complement \left({V}\right) \subseteq U_A$.

From Subset of Union. we also have $A \subseteq A \cup \complement \left({V}\right)$.

Since $\complement \left({A}\right) \cap V$ is open, $A \cup \complement \left({V}\right)$ is closed.

Hence we have found a closed neighborhood for $A$ in $U_A$, as desired.

Definition by Closed Neighborhoods implies Definition by Open Sets
Let $T$ satisfy the definition by closed neighborhoods of a $T_4$ space.

Let $A$ and $B$ be disjoint closed sets in $T$.

Then from Empty Intersection iff Subset of Complement, we have:


 * $A \subseteq \complement \left({B}\right)$

and the latter is open in $T$.

Applying the assumption, we find a closed neighborhood $C_A$ of $A$ contained in $\complement \left({B}\right)$.

From Empty Intersection iff Subset of Complement and Complements Invert Subsets we establish:


 * $A \cap \complement \left({C_A}\right) = \varnothing$
 * $B \subseteq \complement \left({C_A}\right)$

Similarly, we find a closed neighborhood $C_B$ of $B$ contained in $\complement \left({C_A}\right)$.

Then from Intersection of Complement with Subset is Empty:


 * $B \cap \complement \left({C_B}\right) = \varnothing$

But from Complements Invert Subsets, we have:


 * $\complement \left({C_A}\right) \subseteq \complement \left({A}\right)$

and so from Subset Relation is Transitive:


 * $C_B \subseteq \complement \left({A}\right)$

Finally, another application of Complements Invert Subsets shows:


 * $A \subseteq \complement \left({C_B}\right)$

Since $C_B \subseteq \complement \left({C_A}\right)$, Empty Intersection iff Subset of Complement shows that $C_A$ and $C_B$ are disjoint sets.

They are also open sets, being the complement of closed sets.

Above, we established also that:


 * $A \subseteq \complement \left({C_B}\right)$
 * $B \subseteq \complement \left({C_A}\right)$

and hence conclude that $T$ satisfies the definition by open sets as well.