Equivalence of Definitions of T4 Space

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

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 $A$.

By definition of open neighborhood:
 * $A \subseteq U_A$

Let $B := \relcomp S {U_A}$.

By Intersection with Complement is Empty iff Subset:
 * $A \cap B = \O$

As $B$ is the complement of $U_A$ in $S$, it is by definition closed in $T$.

Thus $A$ and $B$ are disjoint closed sets in $T$.

Also, from Set Complement inverts Subsets:
 * $B \subseteq \relcomp S A$

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

From Set Intersection Preserves Subsets:
 * $B \subseteq \map \complement A \cap V$

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

Then, from Complement of Complement, De Morgan's Laws: Complement of Union and Set Complement inverts Subsets:


 * $A \cup \map \complement V \subseteq U_A$

From Subset of Union we also have:
 * $A \subseteq A \cup \map \complement V$

Since $\relcomp S A \cap V$ is open, $A \cup \relcomp S V$ is closed.

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

Hence it is concluded that $T$ satisfies the definition by closed neighborhoods of a $T_4$ space.

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:


 * $A \subseteq \map \complement B$

and the latter is open in $T$.

Applying the assumption, we find a closed neighborhood $C_A$ of $A$ contained in $\map \complement B$.

From Empty Intersection iff Subset of Complement and Set Complement inverts Subsets we establish:


 * $A \cap \map \complement {C_A} = \O$
 * $B \subseteq \map \complement {C_A}$

Similarly, we find a closed neighborhood $C_B$ of $B$ contained in $\map \complement {C_A}$.

Then from Intersection with Complement is Empty iff Subset:


 * $B \cap \map \complement {C_B} = \O$

But from Set Complement inverts Subsets, we have:


 * $\map \complement {C_A} \subseteq \map \complement A$

and so from Subset Relation is Transitive:


 * $C_B \subseteq \map \complement A$

Finally, another application of Set Complement inverts Subsets shows:


 * $A \subseteq \map \complement {C_B}$

Since $C_B \subseteq \map \complement {C_A}$, 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:


 * $A \subseteq \map \complement {C_B}$
 * $B \subseteq \map \complement {C_A}$

Hence it is concluded that $T$ satisfies the definition by open sets as well.