Definition:T4 Space/Definition 2

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

$T = \struct {S, \tau}$ 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.