Separation axioms in topological spaces | |
---|---|
Kolmogorov classification | |
T_{0} | (Kolmogorov) |
T_{1} | (Fréchet) |
T_{2} | (Hausdorff) |
T_{2½} | (Urysohn) |
completely T_{2} | (completely Hausdorff) |
T_{3} | (regular Hausdorff) |
T_{3½} | (Tychonoff) |
T_{4} | (normal Hausdorff) |
T_{5} | (completely normal Hausdorff) |
T_{6} | (perfectly normal Hausdorff) |
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.
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. (The closures themselves do not have to be disjoint.) 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.
For more on these conditions (including their use outside the separation axioms), see Separated sets and Topological distinguishability. |
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 "T_{4}" are sometimes interchanged, similarly "regular" and "T_{3}", 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.
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 T_{1} axiom, then each axiom also implies the ones in the cells above it (for example, all normal T_{1} spaces are also completely regular).
Separated | Separated by neighborhoods | Separated by closed neighborhoods | Separated by function | Precisely separated by function | |
---|---|---|---|---|---|
Distinguishable points | Symmetric^{[4]} | Preregular | |||
Distinct points | Fréchet | Hausdorff | Urysohn | Completely Hausdorff | Perfectly Hausdorff |
Closed set and point outside | Symmetric^{[5]} | Regular | Completely regular | Perfectly normal | |
Disjoint closed sets | always | Normal | |||
Separated sets | always | Completely normal | discrete space |
The T_{0} axiom is special in that it can not only be added to a property (so that completely regular plus T_{0} is Tychonoff) but also be subtracted from a property (so that Hausdorff minus T_{0} is R_{1}), 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 T_{0}, and one goes from the left side to the right side by removing that requirement, using the Kolmogorov quotient operation. (The names in parentheses given on the left side of this table are generally ambiguous or at least less well known; but they are used in the diagram below.)
T_{0} version | Non-T_{0} version |
---|---|
T_{0} | (No requirement) |
T_{1} | R_{0} |
Hausdorff (T_{2}) | R_{1} |
T_{2½} | (No special name) |
Completely Hausdorff | (No special name) |
Regular Hausdorff (T_{3}) | Regular |
Tychonoff (T_{3½}) | Completely regular |
Normal T_{0} | Normal |
Normal Hausdorff (T_{4}) | Normal regular |
Completely normal T_{0} | Completely normal |
Completely normal Hausdorff (T_{5}) | Completely normal regular |
Perfectly normal Hausdorff (T_{6}) | Perfectly normal |
Other than the inclusion or exclusion of T_{0}, the relationships between the separation axioms are indicated in the diagram to the right. In this diagram, the non-T_{0} version of a condition is on the left side of the slash, and the T_{0} version is on the right side. Letters are used for abbreviation as follows: "P" = "perfectly", "C" = "completely", "N" = "normal", and "R" (without a subscript) = "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 ("CN") and completely Hausdorff ("CT_{2}"), then following both branches up, one finds he spot "•/T_{5}". Since completely Hausdorff spaces are T_{0} (even though completely normal spaces may not be), one takes the T_{0} side of the slash, so a completely normal completely Hausdorff space is the same as a T_{5} space (less ambiguously known as a completely normal Hausdorff space, as can be seen in the table above).
As can be seen from the diagram, normal and R_{0} 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 R_{0} are typically called "normal regular spaces". In a somewhat similar fashion, spaces that are both normal and T_{1} 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.
[NB: This diagram does not reflect that perfectly normal spaces are always regular; the editors are working on this now.]
There are some other conditions on topological spaces that are sometimes classified with the separation axioms, but these don't fit in with the usual separation axioms as completely. Other than their definitions, they aren't discussed here; see their individual articles.