# Definition:Fully T4 Space

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## Definition

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

$T$ is **fully $T_4$** if and only if every open cover of $S$ has a star refinement.

## Variants of Name

From about 1970, treatments of this subject started to refer to this as a **fully normal space**, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a fully normal space as a **fully $T_4$ 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 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 T4 Spaces**can be found here.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 3$: Paracompactness