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

• ${\displaystyle \mathbb {Z} }$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. ${\displaystyle x\in \mathbb {Z} }$ ${\displaystyle G_{x}}$ ${\displaystyle G_{x}=X\setminus \{x,x+1\}}$ ${\displaystyle x}$ ${\displaystyle G_{x}=X\setminus \{x-1,x\}}$ ${\displaystyle x}$ ${\displaystyle A}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle U_{A}=\cap _{x\in A}G_{x}}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle x+1}$

• 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 . ${\displaystyle (X,d)}$ ${\displaystyle d}$ ${\displaystyle x\in X}$ ${\displaystyle y\in X}$ ${\displaystyle d(x,y)=0}$