Definition:Fully Normal Space

From ProofWiki
Jump to navigation Jump to search


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 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 fully normal spaces can be found here.