Definition:Completely Hausdorff Space

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

$\struct {S, \tau}$ is a completely Hausdorff space or $T_{2 \frac 1 2}$ space if and only if:

$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U^- \cap V^- = \O$

That is, for any two distinct elements $x, y \in S$ there exist open sets $U, V \in \tau$ containing $x$ and $y$ respectively whose closures are disjoint.

That is:

$\struct {S, \tau}$ is a $T_{2 \frac 1 2}$ space if and only if every two points in $S$ are separated by closed neighborhoods.

Source of Name

This entry was named for Felix Hausdorff.

Variants of Name

From about 1970, treatments of this subject started to refer to this as an Urysohn space, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as an Urysohn space as a completely Hausdorff 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 known as

Some sources give this as $T_{\frac 5 2}$ space, which of course evaluates to the same as a $T_{2 \frac 1 2}$ space.

Also see

  • Results about completely Hausdorff spaces can be found here.