# Equivalence of Definitions of T4 Space

## Contents

## Theorem

The following definitions of the concept of **$T_4$ space** are equivalent:

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

### Definition by Open Sets

$T = \left({S, \tau}\right)$ is a **$T_4$ space** if and only if:

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

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

### Definition by Closed Neighborhoods

$T = \left({S, \tau}\right)$ is **$T_4$** if and only if each open set $U$ contains a closed neighborhood of each closed set contained in $U$.

## Proof

### 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 $B := \complement_S \left({U_A}\right)$.

By Intersection with Complement is Empty iff Subset:

- $A \cap B = \varnothing$

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:

- $B \subseteq \complement_S \left({A}\right)$

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

From Set Intersection Preserves Subsets:

- $B \subseteq \complement \left({A}\right) \cap V$

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

Then, from Complement of Complement, De Morgan's Laws: Complement of Union and Set Complement inverts 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 $\complement_S \left({A}\right) \cap V$ is open, $A \cup \complement_S \left({V}\right)$ 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.

$\Box$

### 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 Set Complement inverts 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 with Complement is Empty iff Subset:

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

But from Set Complement inverts 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 Set Complement inverts 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:

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

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

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
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