Separation Axiom - Preliminary Definitions

Preliminary Definitions

Before we define the separation axioms themselves, we give concrete meaning to the concept of separated sets (and points) in topological spaces. (But separated sets are not the same as separated spaces, defined in the next section.)

The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points. It's not enough for elements of a topological space to be distinct; we may want them to be topologically distinguishable. Similarly, it's not enough for subsets of a topological space to be disjoint; we may want them to be separated (in any of various ways). The separation axioms all say, in one way or another, that points or sets that are distinguishable or separated in some weak sense must also be distinguishable or separated in some stronger sense.

Let X be a topological space. Then two points x and y in X are topologically distinguishable if they do not have exactly the same neighbourhoods (or equivalently the same open neighbourhoods); that is, at least one of them has a neighbourhood that is not a neighbourhood of the other (or equivalently there is an open set that one point belongs to but the other point does not).

Two points x and y are separated if each of them has a neighbourhood that is not a neighbourhood of the other; that is, neither belongs to the other's closure. More generally, two subsets A and B of X are separated if each is disjoint from the other's closure. (The closures themselves do not have to be disjoint.) The points x and y are separated if and only if their singleton sets {x} and {y} are separated; all of the remaining conditions for sets may also be applied to points (or to a point and a set) by using singleton sets.

To continue, subsets A and B are separated by neighbourhoods if they have disjoint neighbourhoods. They are separated by closed neighbourhoods if they have disjoint closed neighbourhoods. They are separated by a function if there exists a continuous function f from the space X to the real line R such that the image f(A) equals {0} and f(B) equals {1}. Finally, they are precisely separated by a function if there exists a continuous function f from X to R such that the preimage f -1({0}) equals A and f -1({1}) equals B.

These conditions are given in order of increasing strength: Any two topologically distinguishable points must be distinct, and any two separated points must be topologically distinguishable. Furthermore, any two separated sets must be disjoint, any two sets separated by neighbourhoods must be separated, and so on.

For more on these conditions (including their use outside the separation axioms), see the articles Separated sets and Topological distinguishability.