Definition:Completely Hausdorff Space

Definition

Let $T = \left({X, \vartheta}\right)$ be a topological space.

$\left({X, \vartheta}\right)$ is a completely Hausdorff space or $T_{2 \frac 1 2}$ space iff:

$\forall x, y \in X, x \ne y: \exists U, V \in \vartheta: x \in U, y \in V: U^- \cap V^- = \varnothing$

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

That is:

$\left({X, \vartheta}\right)$ is a $T_{2 \frac 1 2}$ space iff every two points in $X$ 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 we define 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 separation axioms.

The system as used here broadly follows Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970).

The system used on the Separation axiom page at Wikipedia differs from this.

Also see

• Results about completely Hausdorff spaces can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 2$: Completely Hausdorff Spaces