# Urysohn room

In topology and related areas of mathematics , Urysohn spaces (named after Pavel Urysohn ) are special topological spaces that meet certain properties.

## definition

Be a topological space. We say that two points and are separated by closed neighborhoods if there are disjoint closed neighborhoods of and . ${\ displaystyle X}$${\ displaystyle x}$${\ displaystyle y}$ ${\ displaystyle x}$${\ displaystyle y}$

${\ displaystyle X}$is a Urysohn space , if two different points are separated by closed sets. It is also said that the axiom of separation satisfies T . ${\ displaystyle X}$

## Relationships with the other axioms of separation

Every Urysohn room is a Hausdorff room and thus fulfills the axioms of separation and . ${\ displaystyle T_ {0}, T_ {1}}$${\ displaystyle T_ {2}}$

On the other hand, every regular Hausdorff room as well as every complete Hausdorff room is a Urysohn room.

## example

In the following we construct a topological space that is a Urysohn space, but not a regular space and also not a complete Hausdorff space. Let the set of rational points in the unit square in , without the pairs, with the first coordinate . Let the set be united with the points and and all points , running over all rational numbers . The open quantities are given by the following environmental bases: ${\ displaystyle S}$${\ displaystyle \ mathbb {Q} ^ {2}}$${\ displaystyle {\ frac {1} {2}}}$${\ displaystyle X}$${\ displaystyle S}$${\ displaystyle (0,0)}$${\ displaystyle (1,0)}$${\ displaystyle (1/2, r {\ sqrt {2}})}$${\ displaystyle r}$${\ displaystyle 0

• for the points from those induced by the Euclidean topology,${\ displaystyle S}$
• for the points of the form , where and for all natural numbers together with ,${\ displaystyle (0,0)}$${\ displaystyle (x, y)}$${\ displaystyle 0 ${\ displaystyle 0 ${\ displaystyle n}$${\ displaystyle (0,0)}$
• for the points of the form , where and for all natural numbers , together with ,${\ displaystyle (1,0)}$${\ displaystyle (x, y)}$${\ displaystyle 3/4 ${\ displaystyle 0 ${\ displaystyle n}$${\ displaystyle (1,0)}$
• for the points of the form , where and .${\ displaystyle (1/2, r {\ sqrt {2}})}$${\ displaystyle (x, y)}$${\ displaystyle 1/4 ${\ displaystyle | yr {\ sqrt {2}} | <1 / n}$

## Comment on the designation

In a complete Hausdorff space there is, by definition, a Urysohn function for two different points, so that it would be obvious to exchange the definitions for Urysohn space and complete Hausdorff space . This is exactly what happened in the book Counterexamples in Topology given below . One should therefore check the definitions used by an author.

## Individual evidence

1. Stephen Willard: General Topology. Adison-Wesley-Publ., 1998, ISBN 0-486-43479-6 , exercise 14F.
2. Steven A. Gall: Point Set Topology. Dover Publ., 2009, ISBN 978-0-486-47222-5 , Chapter II.2, p. 83.
3. Michel Coornaert: Topological dimension and Dynamical Systems. Springer-Verlag, 2015, ISBN 978-3-319-19793-7 , chap. I.5, p. 102.
4. ^ JR Porter, RG Woods: Extensions and Absoluteness of Hausdorff Spaces. Springer-Verlag, 1988, ISBN 1-4612-8316-7 , chapter 4.8, p. 305.
5. Lynn Arthur Steen, J. Arthur Seebach: Counterexamples in Topology. Springer-Verlag, 1978, ISBN 3-540-90312-7 , p. 13 and p. 16.