Definition:Completely Hausdorff Space

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

Definition 1

$\struct {S, \tau}$ 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.

Definition 2

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

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

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.

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 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 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.

Source of Name

This entry was named for Felix Hausdorff.