In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff. The separation axioms are not fundamental axioms like those of set theory, but rather defining properties which may be specified to distinguish certain types of topological spaces. The separation axioms are denoted with the letter "T" after the German Trennungsaxiom ("separation axiom"), and increasing numerical subscripts denote stronger and stronger properties. The precise definitions of the separation axioms have varied over time. Especially in older literature, different authors might have different definitions of each condition.
Preliminary definitions Before we define the separation axioms themselves, we give concrete meaning to the concept of separated sets (and points) in topological spaces. (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 (that is, unequal); 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). That is, at least one of the points does not belong to the other's closure. 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, though the closures themselves do not have to be disjoint. Equivalently, each subset is included in an open set disjoint from the other subset. All of the remaining conditions for separation of sets may also be applied to points (or to a point and a set) by using singleton sets. Points x and y will be considered separated, by neighbourhoods, by closed neighbourhoods, by a continuous function, precisely by a function, if and only if their singleton sets {x} and {y} are separated according to the corresponding criterion. 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 continuous function if there exists a continuous function f from the space X to the real line R such that A is a subset of the preimage f−1({0}) and B is a subset of the preimage f−1({1}). Finally, they are precisely separated by a continuous function if there exists a continuous function f from X to R such that A equals the preimage f−1({0}) and B equals f−1({1}). 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. Any two separated sets must be disjoint, any two sets separated by neighbourhoods must be separated, and so on.
Main definitions These definitions all use essentially the preliminary definitions above. Many of these names have alternative meanings in some of mathematical literature; for example, the meanings of "normal" and "T4" are sometimes interchanged, similarly "regular" and "T3", etc. Many of the concepts also have several names; however, the one listed first is always least likely to be ambiguous. Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles. In all of the following definitions, X is again a topological space.
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