R _{0} space
In topology and related areas of mathematics , R _{0} spaces are special topological spaces that have certain pleasant properties. The property of being an R _{0} is counted among the so-called axioms of separation .
definition
A topological space X and two points x and y in X are given . It is said that x and y are or can be separated if x and y are each in an open set that does not contain the other point. Furthermore, x and y are called topologically distinguishable if there is an open set that contains exactly one of the two points.
X is called R _{0} -space if any two topologically distinguishable points are separated. An R _{0} space is also called a symmetric space .
properties
Let X be a topological space. The following statements are equivalent:
- X is an R _{0} space.
- For every x in X , the closure of { x } only contains those points that are topologically indistinguishable from x .
- Each elementary filter for x only converges to points that are topologically indistinguishable from x .
- The Kolmogoroff quotient KQ ( X ) is a T _{1} space.
The following implication always applies in topological spaces
- separated ⇒ topologically distinguishable
If this can be reversed, it is an R _{0} space.
If X is an R _{0} space, this also applies to every subspace .
If ( X _{i} ) is a family of R _{0} -spaces, then its product space is also an R _{0} -space and vice versa.
Examples
- be the set of integers . For is defined by for even and for odd . Passes through the finite subsets of , the sets form a basis of a topology. We received an R _{0} -space, which no Kolmogorov space (for a straight are and topologically indistinguishable) and therefore no T _{1} -space is._{}_{}
- If a pseudometric space is, then this is an R _{0} space with regard to the topology induced by the metric . For the points topologically indistinguishable from a point , just applies ._{}