Separation axiom
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, 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 in topological spaces.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. 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 ; that is, at least one of them has a neighbourhood that is not a neighbourhood of the other. 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 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 and 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 and B is a subset of the preimage f−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 and B equals f−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.X is T0, or Kolmogorov, if any two distinct points in X are topologically distinguishable. X is R0, or symmetric, if any two topologically distinguishable points in X are separated.X is T1, or accessible or Fréchet, if any two distinct points in X are separated. Equivalently, every single-point set is a closed set. Thus, X is T1 if and only if it is both T0 and R0. X is R1, or preregular, if any two topologically distinguishable points in X are separated by neighbourhoods. Every R1 space is also R0.X is Hausdorff, or T2 or separated, if any two distinct points in X are separated by neighbourhoods. Thus, X is Hausdorff if and only if it is both T0 and R1. Every Hausdorff space is also T1.X is completely Hausdorff spaces|T2½], or Urysohn, if any two distinct points in X are separated by closed neighbourhoods. Every T2½ space is also Hausdorff.X is completely Hausdorff, or completely T2, if any two distinct points in X are separated by a continuous function. Every completely Hausdorff space is also T2½.X is regular if, given any point x and closed set F in X such that x does not belong to F, they are separated by neighbourhoods. Every regular space is also R1.X is regular Hausdorff, or T3, if it is both T0 and regular. Every regular Hausdorff space is also T2½.X is completely regular if, given any point x and closed set F in X such that x does not belong to F, they are separated by a continuous function. Every completely regular space is also regular.X is Tychonoff, or T3½, completely T3, or completely regular Hausdorff, if it is both T0 and completely regular. Every Tychonoff space is both regular Hausdorff and completely Hausdorff.X is normal if any two disjoint closed subsets of X are separated by neighbourhoods. X is normal regular if it is both R0 and normal. Every normal regular space is also completely regular.X is normal Hausdorff, or T4, if it is both T1 and normal. Every normal Hausdorff space is also both Tychonoff and normal regular.X is completely normal if any two separated sets are separated by neighbourhoods. Every completely normal space is also normal.X is completely normal Hausdorff, or T5 or completely T4, if it is both completely normal and T1. Every completely normal Hausdorff space is also normal Hausdorff.X is perfectly normal if any two disjoint closed sets are precisely separated by a continuous function. Every perfectly normal space is also both completely normal and completely regular.X is perfectly normal Hausdorff, or T6 or perfectly T4, if it is both perfectly normal and T0. Every perfectly normal Hausdorff space is also completely normal Hausdorff.
The following table summarizes the separation axioms as well as the implications between them: cells which are merged represent equivalent properties, each axiom implies the ones in the cells to its left, and if we assume the T1 axiom, then each axiom also implies the ones in the cells above it.
Relationships between the axioms
The T0 axiom is special in that it can not only be added to a property but also be subtracted from a property, in a fairly precise sense; see Kolmogorov quotient for more information. When applied to the separation axioms, this leads to the relationships in the table to the left below. In this table, one goes from the right side to the left side by adding the requirement of T0, and one goes from the left side to the right side by removing that requirement, using the Kolmogorov quotient operation.[Image:Separation axioms.svg|300px|right|Hasse diagram of the separation axioms.]
| T0 version | Non-T0 version |
| T0 | |
| T1 | R0 |
| Hausdorff | R1 |
| T2½ | |
| Completely Hausdorff | |
| Regular Hausdorff | Regular |
| Tychonoff | Completely regular |
| Normal T0 | Normal |
| Normal Hausdorff | Normal regular |
| Completely normal T0 | Completely normal |
| Completely normal Hausdorff | Completely normal regular |
| Perfectly normal Hausdorff | Perfectly normal |
Other than the inclusion or exclusion of T0, the relationships between the separation axioms are indicated in the diagram to the right. In this diagram, the non-T0 version of a condition is on the left side of the slash, and the T0 version is on the right side. Letters are used for abbreviation as follows:
"P" = "perfectly", "C" = "completely", "N" = "normal", and "R" = "regular". A bullet indicates that there is no special name for a space at that spot. The dash at the bottom indicates no condition.
Two properties may be combined using this diagram by following the diagram upwards until both branches meet. For example, if a space is both completely normal and completely Hausdorff, then following both branches up, one finds the spot "•/T5".
Since completely Hausdorff spaces are T0, one takes the T0 side of the slash, so a completely normal completely Hausdorff space is the same as a T5 space.
As can be seen from the diagram, normal and R0 together imply a host of other properties, since combining the two properties leads through the many nodes on the right-side branch. Since regularity is the most well known of these, spaces that are both normal and R0 are typically called "normal regular spaces". In a somewhat similar fashion, spaces that are both normal and T1 are often called "normal Hausdorff spaces" by people that wish to avoid the ambiguous "T" notation. These conventions can be generalised to other regular spaces and Hausdorff spaces.