Separate quantities

In topology and related fields of mathematics , discrete sets are pairs of subsets of a given topological space that are related in some way. Whether two sets are separate or not is important both for the concept of connected sets and for the axioms of separation for topological spaces.

Definitions

There are different versions of this concept. The terms are defined below. Let X be a topological space.

Two subsets and of are called disjoint if their intersection is empty. This property has nothing to do with topology, but is a concept of set theory . We mention this property here because it is the weakest of the separation properties considered here. See also the article on disjoint sets . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle X}$

${\displaystyle A}$and are called separated in if both sets are disjoint to the closure of the other set. However, it is not required that the two degrees should be disjoint. For example, the intervals and are separated in , although the end of both sets. Furthermore, separate sets are always disjoint. ${\displaystyle B}$ ${\displaystyle X}$ ${\displaystyle \left[0,1\right)}$ ${\displaystyle \left(1,2\right]}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle 1}$

${\displaystyle A}$and are separated by neighborhoods if disjoint neighborhoods of and from exist. In certain books are open environments and demanded. However, this definition is equivalent to the preceding one. For example, and are separated by surroundings, because the and are disjoint surroundings of and . Obviously, sets separated by environments are also separated. ${\displaystyle B}$ ${\displaystyle U}$ ${\displaystyle A}$ ${\displaystyle V}$ ${\displaystyle B}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle A=\left[0,1\right)}$ ${\displaystyle B=\left(1,2\right]}$ ${\displaystyle U=(-1,1)}$ ${\displaystyle V=(1,3)}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle A}$and are separated by closed neighborhoods if disjoint closed neighborhoods of and from exist. The quantities and are not separated by closed environments. By adding we get closed supersets for both sets, but since the closure of both sets, there are no disjoint closed neighborhoods. Further, sets separated by closed environments are also separated by environments. ${\displaystyle B}$ ${\displaystyle U}$ ${\displaystyle A}$ ${\displaystyle V}$ ${\displaystyle B}$ ${\displaystyle \left[0,1\right)}$ ${\displaystyle \left(1,2\right]}$ ${\displaystyle 1}$ ${\displaystyle 1}$

${\displaystyle A}$and are called separated by functions if there is a continuous function of in the real numbers such that and . In the literature, it is sometimes also required that the values ​​be in the interval . However, this definition is equivalent to the above. The two sets and are not separated by functions, because it is not possible to choose the function continuously at the point . Sets separated by functions are also separated by closed environments; The archetypes and for one can be chosen as closed environments . Which spaces fulfill this becomes clear in Urysohn's lemma . ${\displaystyle B}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle f(A)=\left\{0\right\}}$ ${\displaystyle f(B)=\left\{1\right\}}$ ${\displaystyle f}$ ${\displaystyle \left[0,1\right]}$ ${\displaystyle \left[0,1\right)}$ ${\displaystyle \left(1,2\right]}$ ${\displaystyle 1}$ ${\displaystyle U:=f^{-1}(\left[-\epsilon ,\epsilon \right])}$ ${\displaystyle V:=f^{-1}(\left[1-\epsilon ,1+\epsilon \right])}$ ${\displaystyle 0<\epsilon <{\frac {1}{2}}}$

${\displaystyle A}$and are sharply separated by a function if there is a continuous function of in the real numbers such that and . Here, too, it can also be required that his picture has in. Sets sharply separated by functions are also separated by functions. Since and are closed subsets of , only closed sets can be sharply separated by functions. But from the fact that closed sets are separated by functions, it cannot be concluded that they are sharply separated by functions. ${\displaystyle B}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle f^{-1}(0)=A}$ ${\displaystyle f^{-1}(1)=B}$ ${\displaystyle f}$ ${\displaystyle \left[0,1\right]}$ ${\displaystyle \left\{0\right\}}$ ${\displaystyle \left\{1\right\}}$ ${\displaystyle \mathbb {R} }$

Relationship to the axioms of separation and separated spaces

The axioms of separation are conditions which are placed on topological spaces and which can be expressed with the help of the different types of separated sets. The separated topological spaces are exactly those that satisfy the separation axiom T ₂ . More precisely, a topological space is separated if and only if for two different points x and y the single-element sets { x } and { y } are separated by neighborhoods. Such rooms are also called Hausdorff rooms or T ₂ rooms .

Relationship to contiguous spaces

Sometimes it is useful to know for a subset A of a topological space whether it is separated from its own complement . This is certainly true if A is the empty set or the whole space X is. But these are not the only examples. A topological space is said to be connected if the empty set and the entire space X are the only sets that satisfy this property. If a non-empty subset A is separated from its complement, and if the only proper subset of A , which has this property is the empty set, then A one open-connected component of X .

Relationship to topologically distinguishable points

In a topological space X , two points x and y are called topologically distinguishable if there is an open set such that exactly one of the two points belongs to it. For topologically distinguishable points the one-element sets { x } and { y } are disjoint. On the other hand, if the sets { x } and { y } are separate, the points x and y can be topologically distinguished.