Separation axiom

In topology and related areas of mathematics, one often does not consider all topological spaces, but provides certain conditions that should be met by the interest areas. Some of these conditions is called separation axioms or separation properties. They are referred to by Andrei Nikolaevich Tikhonov sometimes called Tikhonov - separation axioms (or in older transcription Tychonoff separation axioms ).

The separation axioms are axioms in the sense that you may require in addition some of these conditions in the definition of a topological space in order to obtain a more restricted notion of topological space. The modern approach is to ( topological space in the article are given as ) the axioms of a topological space to fix once and for all, and then to speak of certain types of topological spaces. The name " separation axiom " for these conditions but has survived to this day. Many separation axioms are denoted by the letter "T" ( for " separation ").

The exact meaning of the terms used in the separation axioms has changed over time. When reading older literature, so you should be sure to know the definition used by the author.

For the formulation of the separation axioms we need some terms that are defined in the following.

Separate volumes and topologically distinguishable points

The separation axioms make statements about how points and levels can be distinguished by topological means. There often is not enough that two points of a topological space are different; one wants to distinguish them topologically. Likewise, it is often not sufficient to allow two sets are disjoint; we want them to be able to separate ( in various ways ) topologically. All require separation axioms that points or amounts which are distinguishable in a certain weak sense, even in a stronger sense are distinguishable.

Let X be a topological space. Two subsets A and B of X are called separated if each of the two disjoint closed to cover the other 's. Separate sets are always disjoint.

There are other, stronger forms of separateness of sets: separated by environments; separated by closed environments; separated by a function; sharply separated by a function. All of these are defined and explained in the article discrete quantities.

Applying the terminology of distinct quantities to points X and Y, we mean the singleton sets { x} {y}. A and B are open disjoint sets, then they are separated by environments: Take U = A and V = B as environments. For this reason, many separation axioms are used specifically on closed sets.

Two points x and y are called topologically distinct if they do not have exactly the same environments. Two topologically distinguishable points are necessarily different. If x and y separately ( ie { x } and { y} are discrete quantities ), then they are topologically distinguishable.

Definition of the separation axioms

Many of the names have changed their meaning over time, and many of these concepts several names. In this encyclopedia, none of these names are often still preferred the order is so arbitrary ( and taken from the English article ).

Most of the axioms can be defined in different ways with the same meaning that given here refers to the first separatory PRIME terms of the previous section.

Be in the following X is always a topological space.

• X is a Kolmogorov space when Axiom T0 met: Any two distinct points of X are topologically distinct, ie there is an open set containing a point, but not the other. Among the other separation axioms, there is often a variation that calls T0, and another that does not.

• X is an R0 - space, or symmetric space if any two topologically distinguishable points are separated, ie when the closed hull of each of the two points does not contain the other.

• X is a T1 - space, or has a Fréchet topology if each two distinct points are separated. The axiom T1 thus consists of T0 and R0. Equivalent to the condition that each singleton is completed. Here to avoid the term " Fréchet space ," which is a term of functional analysis.

• X is a präregulärer space if it satisfies the axiom R1: Two topologically distinct points are separated by environments. The axiom R1 includes R0.

• X is a Hausdorff space if it satisfies the axiom T2: Any two distinct points are separated by environments. The axiom T2 thus consists of T0 and R1. It includes the condition T1. Equivalent to the condition that each have two distinct points have disjoint neighborhoods.

• X is a sober space ( engl. sober space ) if every irreducible closed set degree is exactly one point. Hausdorff spaces are sober and sober spaces are T0.

• X is a Urysohn space if it satisfies the axiom T2 ½: Any two distinct points are separated by closed environments. The axiom T2 ½ includes T2, a Urysohn space is Hausdorff so.

• X is a complete Hausdorff space or completely T2, if any two points separated by a function. Each fully haussdorffsche space is a T2 ½ - room.

• X is a regular space, when each point x, separated from each completed set F, which does not contain x of environments. In a regular space x and F even separated by closed environments. Every regular space is präregulär, and each regular T0- space is Hausdorff.

• X is a regular Hausdorff space shear- or T3 - space if it is T0 and met regularly. A T3 - space is always a T2 ½ - room.

• X is a space completely normal, if each completed set F and is not located in any point F separated by a function. Every completely regular space is regular.

• X is a Tychonoff space or a T3 ½ - room or T3a - room or a complete T3 - space or a completely regular Hausdorff space shear, if X is a T0- space, which is also completely regular. A Tychonoff space is both Hausdorff and completely regular Hausdorff space.

• For the separation axiom T4 and the concept of a normal space, there are two conventions: either T4 states that any two closed disjoint subsets have disjoint open neighborhoods, and a normal space is a space which satisfies T2 and T4
• Or a space is called normal if each have two closed disjoint subsets disjoint open neighborhoods, and a room filled T4 if it is normal and Hausdorff ( ie in addition satisfies T2). This definition we use for the rest of the article.

• X is a completely regular space, if two separate quantities are always separated by environments. A completely normal space is always normal.

• X is a completely regular Hausdorff space or a shear- T5 - room or a complete T4 - space if X is T1 and satisfies both completely normal. Each T5 - space is also a T4 - space.

• X is a perfectly normal space, if two disjoint closed sets are sharply separated by a function. Every perfectly normal space is completely normal.

• X is a perfectly normal Hausdorff space or a perfect shear T4 space if X is perfectly normal and T1 fulfilled. Each room is a perfect T4 - T5 - space.

• X is a locally compact space, if X is Hausdorff and every point has a compact neighborhood.