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 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.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Paracompactness