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 _2½ . ${\ displaystyle X}$

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: ${\ displaystyle S}$ ${\ displaystyle \ mathbb {Q} ^ {2}}$ ${\ displaystyle {\ frac {1} {2}}}$ ${\ displaystyle X}$ ${\ displaystyle S}$ ${\ displaystyle (0,0)}$ ${\ displaystyle (1,0)}$ ${\ displaystyle (1/2, r {\ sqrt {2}})}$ ${\ displaystyle r}$ ${\ displaystyle 0 <r <1 / {\ sqrt {2}}}$

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

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

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

[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.