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.
| Separation axioms in topological spaces | |
|---|---|
| Kolmogorov classification | |
| T0 | (Kolmogorov) |
| T1 | (Fréchet) |
| T2 | (Hausdorff) |
| T2½ | (Urysohn) |
| completely T2 | (completely Hausdorff) |
| T3 | (regular Hausdorff) |
| T3½ | (Tychonoff) |
| T4 | (normal Hausdorff) |
| T5 | (completely normal Hausdorff) |
| T6 | (perfectly normal Hausdorff) |
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
editBefore 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
editThese 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. (It will be a common theme among the separation axioms to have one version of an axiom that requires T0 and one version that doesn't.)
- 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. (Although one may say such things as "T1 space", "Fréchet topology", and "suppose that the topological space X is Fréchet"; one should avoid saying "Fréchet space" in this context, since there is another entirely different notion of Fréchet space in functional analysis.)
- 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 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. (In fact, in a regular space, any such x and F will also be separated by closed neighbourhoods.) Every regular space is also R1.
- X is regular Hausdorff, or T3, if it is both T0 and regular.[1] 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.[2] 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.[3] 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. (In fact, a space is normal if and only if any two disjoint closed sets can be separated by a continuous function; this is Urysohn's lemma.)
- 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 (for example, all normal T1 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 | ||
Relationships between the axioms
editThe T0 axiom is special in that it can not only be added to a property (so that completely regular plus T0 is Tychonoff) but also be subtracted from a property (so that Hausdorff minus T0 is R1), 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. (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.)
| T0 version | Non-T0 version |
|---|---|
| T0 | (No requirement) |
| T1 | R0 |
| Hausdorff (T2) | R1 |
| T2½ | (No special name) |
| Completely Hausdorff | (No special name) |
| Regular Hausdorff (T3) | Regular |
| Tychonoff (T3½) | Completely regular |
| Normal T0 | Normal |
| Normal Hausdorff (T4) | Normal regular |
| Completely normal T0 | Completely normal |
| Completely normal Hausdorff (T5) | Completely normal regular |
| Perfectly normal Hausdorff (T6) | 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" (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 ("CT2"), then following both branches up, one finds the spot "•/T5". Since completely Hausdorff spaces are T0 (even though completely normal spaces may not be), one takes the T0 side of the slash, so a completely normal completely Hausdorff space is the same as a T5 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 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.
[NB: This diagram does not reflect that perfectly normal spaces are always regular; the editors are working on this now.]
Other separation axioms
editThere 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.
- X is sober if, for every closed set C that is not the (possibly nondisjoint) union of two smaller closed sets, there is a unique point p such that the closure of {p} equals C. More briefly, every irreducible closed set has a unique generic point. Any Hausdorff space must be sober, and any sober space must be T0.
- X is weak Hausdorff if, for every continuous map f to X from a compact Hausdorff space, the image of f is closed in X. Any Hausdorff space must be weak Hausdorff, and any weak Hausdorff space must be T1.
- X is semiregular if the regular open sets form a base for the open sets of X. Any regular space must also be semiregular.
- X is quasi-regular if for any nonempty open set G, there is a nonempty open set H such that the closure of H is contained in G.
- X is fully normal if every open cover has an open star refinement. X is fully T4, or fully normal Hausdorff, if it is both T1 and fully normal. Every fully normal space is normal and every fully T4 space is T4. Moreover, one can show that every fully T4 space is paracompact. In fact, fully normal spaces actually have more to do with paracompactness than with the usual separation axioms.
- The axiom that all compact subsets are closed is strictly between T1 and T2 (Hausdorff) in strength. A space satisfying this axiom is necessarily T1 because every single-point set is necessarily compact and thus closed, but the reverse is not necessarily true; for the cofinite topology on infinitely many points, which is T1, every subset is compact but not every subset is closed. Furthermore, every T2 (Hausdorff) space satisfies the axiom that all compact subsets are closed, but the reverse is not necessarily true; for the cocountable topology on uncountably many points, the compact sets are all finite and hence all closed but the space is not T2 (Hausdorff).
See also
editNotes
edit- ^ Schechter 1997, p. 441.
- ^ Schechter 1997, 16.16, p. 442.
- ^ Schechter 1997, 16.17, p. 443.
- ^ Schechter 1997, 16.6(D), p. 438.
- ^ Schechter 1997, 16.6(C), p. 438.
References
edit- Schechter, Eric (1997). Handbook of Analysis and its Foundations. San Diego: Academic Press. ISBN 0126227608. (has Ri axioms, among others)
- Willard, Stephen (1970). General topology. Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0-486-43479-6. (has all of the non-Ri axioms mentioned in the Main Definitions, with these definitions)