# 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}$