Separated sets: Encyclopedia II - Separated sets - Definitions

Separated sets - Definitions

There are various versions of the concept. The terms are defined below, where X is a topological space.

First, two subsets A and B of X 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 don't 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. Note that any two separated sets automatically must be disjoint.

A and B are separated by neighbourhoods if there are a neighbourhood U of A and a neighbourhood 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.

A and B are separated by closed neighbourhoods if there are 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 can't 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 it makes no difference in the end.) 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).

Adapted from the Wikipedia article "Definitions", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki