Definition:Urysohn Space
Definition
Let $T = \struct {S, \tau}$ be a topological space.
$\struct {S, \tau}$ is an Urysohn space if and only if:
- For any distinct elements $x, y \in S$ (that is, $x \ne y$), there exists an Urysohn function for $\set x$ and $\set y$.
Source of Name
This entry was named for Pavel Samuilovich Urysohn.
Variants of Name
From about 1970, treatments of this subject started to refer to this as a completely Hausdorff space, and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a completely Hausdorff space as an Urysohn 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 see
- Results about Urysohn spaces can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Additional Separation Properties