Definition:Urysohn Space
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Definition
Let $T = \left({X, \vartheta}\right)$ be a topological space.
$\left({X, \vartheta}\right)$ is an Urysohn space iff:
- For any distinct points $x, y \in X$ (i.e. $x \ne y$), there exists an Urysohn function for $\left\{{x}\right\}$ and $\left\{{y}\right\}$.
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 we define 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 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 Urysohn spaces can be found here.
Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 2$: Additional Separation Properties
