Separated sets

In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces (and their connected components) as well as to the separation axioms for topological spaces. Separated sets should not be confused with separated spaces (defined below), which are somewhat related but different. Separable spaces are again a completely different topological concept. There are various ways in which two subsets and of a topological space can be considered to be separated. A most basic way in which two sets can be separated is if they are disjoint, that is, if their intersection is the empty set. This property has nothing to do with topology as such, but only set theory. Each of the properties below is stricter than disjointness, incorporating some topological information. The properties are presented in increasing order of specificity, each being a stronger notion than the preceding one. A more restrictive property is that and are in if each is disjoint from the other's closure: This property is known as the . Since every set is contained in its closure, two separated sets automatically must be disjoint. The closures themselves do not have to be disjoint from each other; for example, the intervals and are separated in the real line even though the point 1 belongs to both of their closures. A more general example is that in any metric space, two open balls and are separated whenever The property of being separated can also be expressed in terms of derived set (indicated by the prime symbol): and are separated when they are disjoint and each is disjoint from the other's derived set, that is, (As in the case of the first version of the definition, the derived sets and are not required to be disjoint from each other.) The sets and are if there are neighbourhoods of and of such that and are disjoint.