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 as well as to the separation axioms for topological spaces. Separated sets should not be confused with separated spaces, which are somewhat related but different. Separable spaces are again a completely different topological concept.
Definitions
There are various ways in which two subsets of a topological space X can be considered to be separated.
A and B are separated in X if each is disjoint from the other's closure. The closures themselves do not have to be disjoint from each other; for example, the intervalsneighbourhoods if there are neighbourhoods U of A and V of B such that U and V are disjoint. and B = and V =. Note that if any two sets are separated by neighbourhoods, then certainly they are separated. If A and B are open and disjoint, then they must be separated by neighbourhoods; just take U=A and V=B. For this reason, separatedness is often used with [closed sets">neighbourhood (topology)">neighbourhoods if there are neighbourhoods U of A and V of B such that U and V are disjoint. and B = and V =. Note that if any two sets are separated by neighbourhoods, then certainly they are separated. If A and B are open and disjoint, then they must be separated by neighbourhoods; just take U=A and V=B. For this reason, separatedness is often used with [closed sets.
Relation to separation axioms and separated spaces
The separation axioms are various conditions that are sometimes imposed upon topological spaces, many of which can be described in terms of the various types of separated sets. As an example we will define the T2 axiom, which is the condition imposed on separated spaces. Specifically, a topological space is separated if, given any two distinct points x and y, the singleton sets and are separated by neighbourhoods. Separated spaces are also called Hausdorff spaces or T2 spaces. Further discussion of separated spaces may be found in the articleHausdorff space. General discussion of the various separation axioms is in the article Separation axiom.
Relation to connected spaces
Given a topological space X, it is sometimes useful to consider whether it is possible for a subsetA to be separated from its complement. This is certainly true if A is either the empty set or the entire space X, but there may be other possibilities. A topological space X is connected if these are the only two possibilities. Conversely, if a nonempty subset A is separated from its own complement, and if the only subset of A to share this property is the empty set, then A is an open-connected component of X. For more on connected spaces, see Connected space.
Given a topological space X, two points x and y are topologically distinguishable if there exists an open set that one point belongs to but the other point does not. If x and y are topologically distinguishable, then the singleton sets and must be disjoint. On the other hand, if the singletons and are separated, then the points x and y must be topologically distinguishable. Thus for singletons, topological distinguishability is a condition in between disjointness and separatedness. For more about topologically distinguishable points, see Topological distinguishability.
