Definition:Urysohn Space

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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.