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 axioms only in the sense that, when defining the notion of topological space, one could add these conditions as extra axioms to get a more restricted notion of what a topological space is. The modern approach is to fix once and for all the axiomatization of topological space and then speak of kinds of topological spaces. However, the term "separation axiom" has stuck. The separation axioms are denoted with the letter "T" after the German Trennungsaxiom, which means "separation axiom."
The precise meanings of the terms associated with the separation axioms has varied over time, as explained in History of the separation axioms. It is important to understand the authors' definition of each condition mentioned to know exactly what they mean, especially when reading older literature.
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).
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 the image f(A) equals {0} and f(B) equals {1}. Finally, they are precisely separated by a continuous 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. 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.
Main definitions
These definitions all use essentially the preliminary definitions above.
Many of these names have alternative meanings in some of mathematical literature, as explained on History of the separation axioms; 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.
- X is T_{0}, 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 T_{0} and one version that doesn't.)
- X is R_{0}, or symmetric, if any two topologically distinguishable points in X are separated.
- X is T_{1}, or accessible or Fréchet or Tikhonov, if any two distinct points in X are separated. Equivalently, every single-point set is a closed set. Thus, X is T_{1} if and only if it is both T_{0} and R_{0}. (Although you may say such things as "T_{1} space", "Fréchet topology", and "suppose that the topological space X is Fréchet"; avoid saying "Fréchet space" in this context, since there is another entirely different notion of Fréchet space in functional analysis.)
- X is R_{1}, or preregular, if any two topologically distinguishable points in X are separated by neighbourhoods. Every R_{1} space is also R_{0}.
- X is Hausdorff, or T_{2} or separated, if any two distinct points in X are separated by neighbourhoods. Thus, X is Hausdorff if and only if it is both T_{0} and R_{1}. Every Hausdorff space is also T_{1}.
- X is T_{2½}, or Urysohn, if any two distinct points in X are separated by closed neighbourhoods. Every T_{2½} space is also Hausdorff.
- X is completely Hausdorff, or completely T_{2}, if any two distinct points in X are separated by a continuous function. Every completely Hausdorff space is also T_{2½}.
- 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 R_{1}.
- X is regular Hausdorff, or T_{3}, if it is both T_{0} and regular.^{[1]} Every regular Hausdorff space is also T_{2½}.
- 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 T_{3½}, completely T_{3}, or completely regular Hausdorff, if it is both T_{0} and completely regular.^{[2]} 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 R_{0} and normal. Every normal regular space is regular.
- X is normal Hausdorff, or T_{4}, if it is both T_{1} and normal. Every normal Hausdorff space is 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 T_{5} or completely T_{4}, if it is both completely normal and T_{1}. 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 completely normal.
- X is perfectly normal Hausdorff, or T_{6} or perfectly T_{4}, if it is both perfectly normal and T_{1}. 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 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 | Preregular | |||
Distinct Points | Fréchet | Hausdorff | Urysohn | Completely Hausdorff | Perfectly Hausdorff |
Closed Set and Point Outside | (always true) | Regular | Completely regular | Perfectly regular | |
Disjoint Closed Sets | (always true) | Normal | Perfectly normal | ||
Separated Sets | (always true) | Completely normal |
Relationships between the axioms
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, you go from the right side to the left side by adding the requirement of T_{0}, and you go 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 T_{0} | Perfectly normal |
Perfectly normal Hausdorff (T_{6}) | Perfectly normal regular |
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.
You can combine two properties 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, you find the spot "•/T_{5}". Since completely Hausdorff spaces are T_{0} (even though completely normal spaces may not be), you take 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 you can see in the table above).
As you can see from the diagram, normal and R_{0} together imply a host of other properties, since combining the two properties leads you to follow a path through the many nodes on the rightside 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.
Other separation axioms
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.
- 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 T_{0}.
- 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 T_{1}.
- 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 T_{4}, or fully normal Hausdorff, if it is both T_{1} and fully normal. Every fully normal space is normal and every fully T_{4} space is T_{4}. Moreover, one can show that every fully T_{4} 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 T_{1} and T_{2} (Hausdorff) in strength. A space satisfying this axiom is necessarily T_{1} 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 T_{1}, every subset is compact but not every subset is closed. Furthermore, every T_{2} (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 T_{2} (Hausdorff).
See also
Notes
References
- Schechter, Eric (1997). Handbook of Analysis and its Foundations. San Diego: Academic Press. ISBN 0126227608. (has R_{i} axioms, among others)
- Willard, Stephen (1970). General topology. Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0-486-43479-6. (has all of the non-R_{i} axioms mentioned in the Main Definitions, with these definitions)
- Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: Wiley. ISBN 0-471-83817-9. (gives a readable introduction to the separation axioms with an emphasis on finite spaces)