Separation axiom

In topology and related areas of mathematics , one often does not consider all topological spaces , but rather sets certain conditions that should be met by the spaces of interest. Some of these conditions are called axioms of separation or separation properties . After Andrei Nikolajewitsch Tichonow they are sometimes also referred to as Tichonow separation axioms (or in an older transcription Tychonoff separation axioms ).

The axioms of separation are axioms in the sense that when defining a topological space, some of these conditions can be additionally required in order to obtain a more restricted concept of topological space. The modern approach consists in fixing the axioms of topological space once and for all (as given in the article on topological space ) and then speaking of certain kinds of topological spaces. The name “separation axiom” for these conditions has been preserved to this day. Many axioms of separation are denoted by the letter "T" (for "separation").

The exact meaning of the terms that appear in the axioms of separation has changed over time. So when reading older literature, one should be careful to know the definition used by the author.

To formulate the axioms of separation we need some terms, which are defined below.

Separate sets and topologically distinguishable points

The axioms of separation make statements about how points and sets can be distinguished using topological means. It is often not enough that two points in a topological space are different; one wants to be able to distinguish them topologically . Likewise, it is often not enough that two sets are disjoint ; we want to be able to separate them topologically (in the most varied of ways) . All axioms of separation require that points or sets that are distinguishable in a certain weak sense are also distinguishable in a stronger sense.

Be a topological space. Two subsets and separated from hot if each of the two is disjoint to the closed hull of the other. Separate sets are always disjoint. ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle X}$

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

If one applies the terminology of separate sets to points and , one means the one-element sets . If and are open disjoint sets, then they are separated by surroundings: take and as surroundings. For this reason, many axioms of separation apply specifically to closed sets. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \left\{x\right\},\left\{y\right\}}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle U=A}$ ${\displaystyle V=B}$

Two points and are called topologically distinguishable if they do not have exactly the same surroundings. Two topologically distinguishable points are necessarily different. If and are separate (that is, and are separate sets), then they are topologically distinguishable. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \left\{x\right\}}$ ${\displaystyle \left\{y\right\}}$

Definition of the axioms of separation

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

Most of the axioms can be defined in different ways with the same meaning, the one given here first referring to the concepts of separation in the previous section.

In the following, always be a topological space. ${\displaystyle X}$

• ${\displaystyle X}$is a Kolmogoroff space if it fulfills the axiom T ₀ : Any two different points of are topologically distinguishable, i. that is, there is an open set that contains one point but not the other. Among the other axioms of separation, there is often a variant that requires T ₀ and another that does not. ${\displaystyle X}$

• ${\displaystyle X}$is an R ₀ -space , or symmetrical space , if two topologically distinguishable points are separated, i.e. if the closed envelope of each of the two points does not contain the other.

• ${\displaystyle X}$is a T ₁ space , or has a Fréchet topology , if two different points are separated. The axiom T ₁ therefore consists of T ₀ and R ₀ . The condition that every single-element set is closed is equivalent . The term " Fréchet room ", which is a term used in functional analysis , should be avoided here .

• ${\displaystyle X}$is a pre-regular space if it fulfills the axiom R ₁ : Two topologically distinguishable points are separated by neighborhoods. The axiom R ₁ includes R ₀ .

• ${\displaystyle X}$is a Hausdorff space if it fulfills the axiom T ₂ : Two different points are separated by neighborhoods. The axiom T ₂ therefore consists of T ₀ and R ₁ . It includes the condition T ₁ . The condition that two different points have disjoint neighborhoods is equivalent.

• ${\displaystyle X}$is a sober space (engl. sober space ), if any irreducible closed set degree is exactly one dot. Hausdorff rooms are sober and sober rooms are T ₀ .

• ${\displaystyle X}$is a Urysohn space if it fulfills the axiom T _2½ : Two different points are separated by closed neighborhoods. The axiom T _2½ includes T ₂ , so a Urysohn space is Hausdorffian.

• ${\displaystyle X}$is a complete Hausdorff space or also complete T ₂ , if any two points are separated by a function. Every completely Haussdorffian room is a T _2½ room.

• ${\displaystyle X}$is a regular space if every point of every closed set that does not contain is separated by neighborhoods. In a regular room, and even separated by closed environments. Every regular room is pre-regular, and every regular T ₀ room is Hausdorffian. ${\displaystyle x}$ ${\displaystyle F}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle F}$

• ${\displaystyle X}$is a regular Hausdorff space or also T ₃ space , if it satisfies T ₀ and is regular. A T ₃ space is always a T _2½ space.

• ${\displaystyle X}$is a completely regular space if every closed set and every point not lying in is separated by a continuous function. Any fully regular room is regular. ${\displaystyle F}$ ${\displaystyle F}$

• ${\displaystyle X}$is a Tychonoff space or a T _3½ space or a T _3a space or a complete T ₃ space or a completely regular Hausdorff space if it is a T ₀ space that is also completely regular. A Tychonoff room is both regular Hausdorffsch and completely Hausdorffsch. ${\displaystyle X}$

• There are two conventions for the axiom of separation T ₄ and the concept of normal space :
  • either T ₄ says that every two closed disjoint subsets have disjoint open neighborhoods, and a normal space is a space that satisfies T ₂ and T ₄
  • or a space is called normal if each two closed disjoint subsets have disjoint open neighborhoods, and a space satisfies T ₄ if it is normal and Hausdorffian (i.e. also satisfies T ₂ ). We'll use this definition for the rest of the article.

• ${\displaystyle X}$is a completely normal space if two separate sets are always separated by surroundings. A perfectly normal room is always normal.

• ${\displaystyle X}$is a completely normal Hausdorff space or a T ₅ space or a complete T ₄ space if both is completely normal and T _{1 is} satisfied. Every T ₅ space is also a T ₄ space. ${\displaystyle X}$

• ${\displaystyle X}$is a perfectly normal space if two disjoint closed sets are sharply separated by a function. Any perfectly normal room is completely normal.

• ${\displaystyle X}$is a perfectly normal Hausdorff room or a perfect T ₄ room if is perfectly normal and fulfills T ₁ . Every perfect T ₄ room is a T ₅ room. ${\displaystyle X}$

• ${\displaystyle X}$is a locally compact space if Hausdorff is and every point has a compact neighborhood. ${\displaystyle X}$

literature

• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .

Individual evidence

1. ↑ Gerhard Wilke: A marking of topological spaces through completions. In: Mathematical Journal . Vol. 182, No. 3, 1983, pp. 339-350, here p. 341, doi : 10.1007 / BF01179754 .