Definition:T4 Space
Definition
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 a $T_4$ space 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.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): normal topological space