Space R0

In Topology, a symmetrical space (or space R₀ ) is a topological particular case of Espace. It is about an example of Axiome of separation.

Definition

That is to say E a topological Space. E is space R₀ so for any couple of elements topologically distinct X and there of E (i.e. there exists a Voisinage one which does not contain the other), there exists an open container X and not there and open containing there and not X.

Properties

That is to say E a topological space. The following properties are equivalent:

E is a R₀ space.
For any X of E, the closing of {X} contains only the points from which X is not topologically distinct.
the principal Ultrafiltre in X converges only towards the points from which X is not topologically distinct.
the Quotient of Kolmogorov of E is T₁.
All open is the union of closed.

A R₀ space which is also T₀ is T₁.

Examples

Is $\ mathbb Z$ the whole of the natural whole . For all $x \ in \ mathbb {Z}$ , one defines $G_x$ such as $G_x = \ mathbb Z \ setminus \ {X, x+1 \}$ if X is even and $G_x = X \ setminus \ {x-1, X \}$ if X is odd. The whole of the $G_x$ defines a Prébase on $\ mathbb Z$ ; a bases can be built by considering the finished intersections of these subsets: the whole of open of $U_A type = \ bigcap_ {X \ in has} G_x$ where $A$ is a finished subset of $\ mathbb Z$ define a topology $\ mathbb Z$ . Topological space thus created is R₀; it is not on the other hand T₀ (and thus not T₁).

See too

Axiom of separation

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