Separated Sets - Definitions

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; we include it here because it is the weakest in the sequence of different notions. For more on disjointness in general, see: disjoint sets.

• 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 are separated in the real line R, even though the point 1 belongs to both of their closures. More generally 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 =, 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, 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 used in place of R in this definition, but it makes no difference in the end.) In our example, 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 and V := f-1, 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).