Separated sets
In topology and related areas of mathematics are discrete quantities pairs of subsets of a given topological space, which are in a way related to each other. The fact that two sets are separated or not, both the concept of related quantities, as well as for the separation of spaces of axioms for topological significance.
Definitions
There are different versions of this concept. The terms are defined below. Here, let X be a topological space.
Two subsets A and B of X are called disjoint if their intersection is empty. This property has nothing to do with the topology, but is a term of set theory. We mention this property here, since it is the weakest of the considered separation properties. See also the articles on disjoint sets.
A and B are separated in X if both sets are disjoint at the end of the other set. However, it is not required that the two statements should be disjoint. For example, the interval [0,1 ) and ( 1,2] separately, although one part of the end of both sets. Next discrete quantities are always disjoint.
A and B are separated by environments if disjoint environment U of A and V B exist. In certain books open neighborhoods U and V are required. This definition is equivalent to the foregoing. For example, = [0,1) and B = ( 1,2] separated by environments, because U = ( -1,1 ) A and V = (1,3) are disjoint neighborhoods of A and B. Obviously,, amounts, which are separated by separately environments.
A and B are separated by closed environments, if disjoint closed environments U of A and V B exist. The amounts of [0,1) and ( 1,2] are not separated by closed environments. Adding 1 we get, although for both volumes completed supersets, but there is one in the financial statements of both sets, there are no disjoint closed environments. Next are discrete quantities separated by closed environments by environments.
A and B are separated by functions if a continuous function f exists from X into the real numbers such that f (A) = { } and f ( B) = { 1}. In the literature is sometimes required in addition to the f assumes values in the interval [ 0,1]. This definition is equivalent to the above. The two quantities of [0,1 ) and ( 1,2] are not separated by functions, as it is not possible, the function at the point 1 to continuously select quantities separated by functions are separated by closed environments; . Than closed environments, the preimages U: = f-1 ( [- ε, ε ] ) and V:. = f -1 ( [1- ε, 1 ε ] ) for 0 < ε < ½ which areas fulfill this becomes clear in Lemma Urysohn.
A and B are sharply separated by a function, if a continuous function f from X exists in the real numbers such that f -1 ( 0) = A and f -1 ( 1) = B. Here, too, may be additionally required, that f his image in [0,1] has. Sharply separated by functions quantities are also separated by functions. Since { 0} and { 1} are closed subsets of, only closed sets can be separated by sharp features. However, from the fact that closed sets are separated by functions that can not be concluded that they are sharply separated by functions.
Relationship to the separation axioms and separated spaces
The separation axioms are conditions that are placed on topological spaces and can be expressed with the help of various types of distinct quantities. Thus, the separated topological spaces are precisely those which satisfy the Trennunsaxiom T2. More specifically, a topological space is then separated, if for any two points x and y are singleton sets { x } and { y} are separated by environments. Such spaces are called Hausdorff spaces or T2 spaces.
Relationship to adjoining rooms
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. But these are not the only examples. A topological space is called connected if the empty set and the whole space X are the only quantities 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, the empty set, then A is an open - connected component of X.
Relationship with topologically distinguishable points
In a topological space X, two points are called x and y are topologically distinguishable if there exists an open set such that exactly one of the points belongs to her. For topologically distinguishable points that singleton sets { x } and { y} are disjoint. Andererseit the sets {x} and {y} separated, the points x and y are topologically distinct.
- Set topology