Definition:T4 Space/Definition 2

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

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

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.