Definition:T4 Space/Definition 1

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Definition

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


$T = \left({S, \tau}\right)$ 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.


That is:

$T = \left({S, \tau}\right)$ is $T_4$ when any two disjoint closed subsets of $S$ are separated by neighborhoods.


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 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology.

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