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.
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 following properties is stricter than disjointness, incorporating some topological information.
The properties below are presented in increasing order of specificity, each being a stronger notion than the preceding one.
The closures themselves do not have to be disjoint from each other; for example, the intervals
A more general example is that in any metric space, two open balls
The property of being separated can also be expressed in terms of derived set (indicated by the prime symbol):
(As in the case of the first version of the definition, the derived sets
be open neighbourhoods, but this makes no difference in the end.)
are open and disjoint, then they must be separated by neighbourhoods; just take
For this reason, separatedness is often used with closed sets (as in the normal separation axiom).
are not separated by a function, because there is no way to continuously define
[2] If two sets are separated by a continuous function, then they are also separated by closed neighbourhoods; the neighbourhoods can be given in terms of the preimage of
are precisely separated by a continuous function if there exists a continuous function
only closed sets are capable of being precisely separated by a function, but just because two sets are closed and separated by a function does not mean that they are automatically precisely separated by a function (even a different function).
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 {x} and {y} are separated by neighbourhoods.
Given a topological space X, it is sometimes useful to consider whether it is possible for a subset A 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.
(In the degenerate case where X is itself the empty set
If x and y are topologically distinguishable, then the singleton sets {x} and {y} must be disjoint.
On the other hand, if the singletons {x} and {y} 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.