There are various ways in which two subsets and of a topological space can be considered to be separated. 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 properties below is stricter than disjointness, incorporating some topological information. The properties are presented in increasing order of specificity, each being a stronger notion than the preceding one.
A more restrictive property is that and are separated in if each is disjoint from the other's closure:
This property is known as the Hausdorff−Lennes Separation Condition.[1] Since every set is contained in its closure, two separated sets automatically must be disjoint. The closures themselves do not have to be disjoint from each other; for example, the intervals and are separated in the real line even though the point 1 belongs to both of their closures. A more general example is that in any metric space, two open balls and are separated whenever The property of being separated can also be expressed in terms of derived set (indicated by the prime symbol): and are separated when they are disjoint and each is disjoint from the other's derived set, that is, (As in the case of the first version of the definition, the derived sets and are not required to be disjoint from each other.)
The sets and are separated by neighbourhoods if there are neighbourhoods of and of such that and are disjoint. (Sometimes you will see the requirement that and be open neighbourhoods, but this makes no difference in the end.) For the example of and you could take and Note that if any two sets are separated by neighbourhoods, then certainly they are separated. If and are open and disjoint, then they must be separated by neighbourhoods; just take and For this reason, separatedness is often used with closed sets (as in the normal separation axiom).
The sets and are separated by closed neighbourhoods if there is a closed neighbourhood of and a closed neighbourhood of such that and are disjoint. Our examples, and are not separated by closed neighbourhoods. You could make either or 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.
The sets and are separated by a continuous function if there exists a continuous function from the space to the real line such that and , that is, members of map to 0 and members of map to 1. (Sometimes the unit interval is used in place of in this definition, but this makes no difference.) In our example, and are not separated by a function, because there is no way to continuously define at the point 1.[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 as and where is any positive real number less than
The sets and are precisely separated by a continuous function if there exists a continuous function such that and (Again, you may also see the unit interval in place of and again it makes no difference.) Note that if any two sets are precisely separated by a function, then they are separated by a function. Since and are closed in 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).