Definition:T4 Space

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

Definition 1

$T = \struct {S, \tau}$ is a $T_4$ space if and only if:

$\forall A, B \in \map \complement \tau, A \cap B = \O: \exists U, V \in \tau: A \subseteq U, B \subseteq V, U \cap V = \O$

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 2

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

Variants of Name

From about 1970, treatments of this subject started to refer to this as a normal space, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a normal space as a $T_4$ space.

However, the names are to a fair extent arbitrary and a matter of taste, as there appears to be no completely satisfactory system for naming all these various Tychonoff separation axioms.

The system as used here broadly follows 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.).

The system used on the Separation axiom page at Wikipedia differs from this.

Also see

  • Results about $T_4$ spaces can be found here.