# 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