Separated sets

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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. 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.

Contents

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 [0,1) and (1,2] 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 Br(x) = {y:d(x,y)<r} and Bs(x) = {y:d(x,y)<s} are separated whenever d(x,y) ≥ 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.

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