Separation Properties

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.

Definitions

There are various ways in which two subsets of a topological space X can be considered to be separated.

• A and B are disjoint if their intersection is the empty set. This property has nothing to do with topology as such, but only set theory. It is included here because it is the weakest in the sequence of different notions.
  • 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 intervals [0, 1) and (1, 2] are separated in the real line R, even though the point 1 belongs to both of their closures. A more general example is that in any metric space, two open balls B_r(x₁) = {y : d(x₁, y) < r} and B_s(x₂) = {y : d(x₂, y) < s} are separated whenever d(x₁, x₂) ≥ r + s. Note that any two separated sets automatically must be disjoint.
  • A and B are separated by neighbourhoods if there are neighbourhoods U of A and V of B such that U and V are disjoint. (Sometimes you will see the requirement that U and V be open neighbourhoods, but this makes no difference in the end.) For the example of A = [0, 1) and B = (1, 2], you could take U = (−1, 1) and V = (1, 3). 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 (as in the normal separation axiom).
    • A and B are separated by closed neighbourhoods if there is a closed neighbourhood U of A and a closed neighbourhood V of B such that U and V are disjoint. Our examples, [0, 1) and (1, 2], are not separated by closed neighbourhoods. You could make either U or V closed by including the point 1 in it, but you cannot make them both closed while keeping them disjoint. Note that if any two sets are separated by closed neighbourhoods, then certainly they are separated by neighbourhoods.
      • A and B are separated by a function if there exists a continuous function f from the space X to the real line R such that f(A) = {0} and f(B) = {1}. (Sometimes you will see the unit interval [0,1] used in place of R in this definition, but this makes no difference.) In our example, [0, 1) and (1, 2] are not separated by a function, because there is no way to continuously define f at the point 1. Note that if any two sets are separated by a function, then they are also separated by closed neighbourhoods; the neighbourhoods can be given in terms of the preimage of f as U := f⁻¹[−e, e] and V := f⁻¹[1 − e, 1 + e], as long as e is a positive real number less than 1/2.
        • A and B are precisely separated by a function if there exists a continuous function f from X to R such that f⁻¹(0) = A and f⁻¹(1) = B. (Again, you may also see the unit interval in place of R, and again it makes no difference.) Note that if any two sets are precisely separated by a function, then certainly they are separated by a function. Since {0} and {1} are closed in R, 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).

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 {x} and {y} are separated by neighbourhoods.

Separated spaces are also called Hausdorff spaces or T₂ spaces.

Relation to connected spaces

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 $\emptyset$, authorities differ on whether $\emptyset$ is connected and whether $\emptyset$ is an open-connected component of itself.)

Relation to topologically distinguishable points

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 {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.