Pre-regular space

In topology and related areas of mathematics , pre-regular spaces are special topological spaces that have certain separation properties. They satisfy the axiom of separation R ₁ .

definition

Let X be a topological space . Two points x and y in X are called topologically distinguishable if there is an open set that contains exactly one of the two points. They are also called separated by neighborhoods if they have disjoint open neighborhoods .

X is called pre-regular space if any two topologically distinguishable points are separated by neighborhoods.

Since two points x and y in X are topologically distinguishable if and only if , one can also define: ${\ displaystyle {\ overline {\ {x \}}} \ not = {\ overline {\ {y \}}}}$

X is called a pre-regular space if two points x and y with each have disjoint neighborhoods. ${\ displaystyle {\ overline {\ {x \}}} \ not = {\ overline {\ {y \}}}}$

Pre-regular spaces are also called R ₁ spaces . It is also said that they satisfy the axiom of separation R ₁ .

Examples

Hausdorff spaces are pre-regular, because in them there is separability for every pair of different points, especially for topologically distinguishable points.
A space with the trivial topology is pre-regular, because in it there are no topologically distinguishable points for which a separability property must exist.
The space with the topology is not pre-regular, because the points 0 and 1 are topologically distinguishable by means of the open set , but they cannot be separated by surroundings. ${\ displaystyle X = \ {0.1 \}}$ ${\ displaystyle \ {\ emptyset, \ {0 \}, X \}}$ ${\ displaystyle \ {0 \}}$

properties

A topological space is pre-regular if and only if the Kolmogoroff quotient KQ ( X ) of X is a Hausdorff space .

If a pre- regular space also fulfills compactness conditions , then it fulfills much stronger axioms of separation: For example, every pre- regular locally compact space is completely regular . Compact pre- regular spaces are even normal . Note that some authors use the terms compact , locally compact , fully regular and normal only for Hausdorff spaces.

Individual evidence

↑ Cyrus F. Nourani: A Functorial Model Theory: Newer Applications to Algebraic Topology, Descriptive Sets, and Computing Categories Topos , Apple Academic Press (2014), ISBN 978-1-926895-92-5 , Chapter 7.2
↑ Ladislav Bican, T. Kepka, P. Němec: Rings, Modules, and Preradicals , M. Dekker (1982), ISBN 0-824-71568-3 , 2.36

[1] Cyrus F. Nourani: A Functorial Model Theory: Newer Applications to Algebraic Topology, Descriptive Sets, and Computing Categories Topos , Apple Academic Press (2014), ISBN 978-1-926895-92-5 , Chapter 7.2

[2] Ladislav Bican, T. Kepka, P. Němec: Rings, Modules, and Preradicals , M. Dekker (1982), ISBN 0-824-71568-3 , 2.36