Definition:Fully Normal Space

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Definition

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


$T$ is fully normal if and only if:

Every open cover of $S$ has a star refinement
All points of $T$ are closed.


That is, $T$ is fully normal if and only if:

$T$ is fully $T_4$
$T$ is a $T_1$ (Fréchet) space.


Variants of Name

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


Sources