# 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.