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 axioms only in the sense that, when defining the notion of topological space, you could throw these conditions in 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 "Trennung", which means separation.

The precise meanings of the terms associated with the separation axioms has varied over time, as explained in History of the separation axioms[?]. Especially when reading older literature, be sure to get the authors' definition of each condition mentioned to make sure that you know exactly what they mean.

Before we define the spaces described by the separation axioms, we need to define some terminology in order to give concrete meaning to the concept of separation.

Table of contents 1 Separated sets and topologically distinguishable points 2 Definitions of the axioms 3 Relationships between the axioms 4 Other separation axioms 5 Sources

Separated sets and topologically distinguishable points

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 (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 separated in some stronger sense.

The terms are defined below, where X is a topological space. Note that we sometimes use the terminology of separated sets to refer to points; in that situation, we're really talking about the singleton {x} rather than the point x.

First, two points x and y in X are topologically distinguishable if they don't have exactly the same neighbourhoods. If x belongs to the closure of {y} and y belongs to the closure of {x}, then x and y are topologically indistinguishable; otherwise, they're topologically distinguishable. For example, in an indiscrete space[?], any two points are topologically indistinguishable. Note that if two points are topologically distinguishable, then certainly they are distinct.

Two subsets A and B of X are separated if each is disjoint from the other's closure. That is, A and B are separated if

Cl(A) ∩ B = A ∩ Cl(B) = {};

where Cl indicates closure and {} is the empty set. The closures themselves don't have to be disjoint from each other; for example, the intervals [0,1) and (1,2] are separated in the real line R, even though the point 1 belongs to both of their closures. Note that any two separated sets must be disjoint. Furthermore, if the points x and y are separated (that is if the sets {x} and {y} are separated), then the points are also topologically distinguishable.

A and B are separated by neighbourhoods if there are a neighbourhood U of A and a neighbourhood V of B such that U and V are disjoint. (Sometimes you will see the requirement that U and V be open neighbourhoods, but this makes no difference in the end.) For the example of A = [0,1) and B = (1,2], you could take U = (-1,1) and V = (1,3). Note that if any two sets are separated by neighbourhoods, then certainly they are separated. If A and B are open and disjoint, then they must be separated by neighbourhoods; just take U := A and V := B. For this reason, many separation axioms refer specifically to closed sets.

A and B are separated by closed neighbourhoods if there are a closed neighbourhood U of A and a closed neighbourhood V of B such that U and V are disjoint. Our examples, [0,1) and (1,2], are not separated by closed neighbourhoods. Note that if any two sets are separated by closed neighbourhoods, then certainly they are separated by neighbourhoods.

A and B are separated by a function if there exists a continuous function f from the space X to the real line R such that f(A) = {0} and f(B) = {1}. (Sometimes you will see the unit interval [0,1] used in place of R in this definition, but it makes no difference in the end.) In our example, [0,1) and (1,2] are not separated by a function, because there is no way to continuously define f at the point 1. Note that if any two sets are separated by a function, then they are also separated by closed neighbourhoods.

A and B are precisely separated by a function if there exists a continuous function f from X to R such that f^-1(0) = A and f^-1(1) = B, where f^-1 indicates the preimage. (Again, you may also see the unit interval in place of R, and again it makes no difference.) Since {0} and {1} are closed in R, only closed sets are capable of being precisely separated by a function. Note that if any two sets are precisely separated by a function, then certainly they are separated by a function.

Separated sets also have a relationship with connected spaces, which is not explored in this article. Don't confuse separated sets with separated spaces, which are defined below. Finally, separable spaces are something else entirely.

Definitions of the axioms

Many of these names have alternative meanings in some of mathematical literature, as explained on History of the separation axioms[?]. Many of the concepts also have several names; the one listed first is preferred in Wikipedia.

Most of these axioms have alternative definitions with the same meaning; the definitions given here are those which fall into a consistent pattern relating 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 defintions, 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. Thus, X is T₁ if and only if it is both T₀ and R₀. Although you may say such things as "T₁ space" 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 preregular, or R₁, if any two topologically distinguishable points in X are separated by neighbourhoods. Note that an R₁ space must also be 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. Note that a Hausdorff space must also be T₁.

X is T_2½[?], or Urysohn, if any two distinct points in X are separated by closed neighbourhoods. Note that a T_2½ space must also be Hausdorff.

X is completely Hausdorff[?], or completely T₂, if any two distinct points in X are separated by a function. Note that a completely Hausdorff space must also be T_2½.

X is regular if, given any point x and closed set F in X, if x does not belong to F, then they are separated by neighbourhoods. In fact, in a regular space, any such x and F will also be separated by closed neighbourhoods. Note that a regular space must also be R₁.

X is regular Hausdorff, or T₃, if it is both T₀ and regular. Note that a regular Hausdorff space must also be T_2½.

X is completely regular if, given any point x and closed set F in X, if x does not belong to F, then they are separated by a function. Note that a completely regular space must also be regular.

X is Tychonoff, or T_3½, completely T₃, or completely regular Hausdorff, if it is both T₀ and completely regular. Note that a Tychonoff space must also be both regular Hausdorff and completely Hausdorff.

X is normal if any two disjoint closed subsets of X are separated by neighbourhoods. In fact, in a normal space, any two disjoint sets will also be separated by a function; this is Urysohn's Lemma.

X is normal Hausdorff, or T₄, if it is both T₁ and normal. Note that a normal Hausdorff space must also be both Tychonoff and normal regular.

X is completely normal if any two separated sets are separated by neighbourhoods. Note that a completely normal space must also be normal.

X is completely normal Hausdorff, or T₅ or completely T₄, if it is both completely normal and T₁. Note that a T₅ space must also be T₄.

X is perfectly normal if any two disjoint closed sets are precisely separated by a function. A perfectly normal space must also be completely normal.

X is perfectly normal Hausdorff, or perfectly T₄, if it is both perfectly normal and T₁. A perfectly T₄ space must also be T₅.

Relationships between the axioms

The T₀ axiom is special in that it cannot only be added to a property (so that regular plus T₀ is T₃) but also subtracted from a property (so that Hausdorff minus T₀ is preregular), 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 below:

T0 version	Non-T0 version
T0	No requirement
T1	R0
Hausdorff	Preregular
T2½	No special name
Completely Hausdorff	No special name
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 T0	Perfectly normal
Perfectly normal Hausdorff	Perfectly normal regular

In this table, you go from the right column to the left column by adding the requirement of T₀, and you go from the left column to the right column by removing that requirment, using the Kolmogorov quotient operation.

Other than the inclusion or exclusion of T₀, the relationships between the separation axioms are indicated in the following diagram:

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.)

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 ("CT₂"), then following both branches up, you find the spot "•/T₅". Since completely Hausdorff spaces are T₀ (even though completely normal spaces may not be), you take the T₀ side of the slash, so a completely normal completely Hausdorff space is the same as a T₅ space.

As you can see from the diagram, normal and R₀ together imply a host of other properties. Since regularity is the most well known of these, spaces that are both normal and R₀ are typically called "normal regular spaces" by people that wish to avoid the "R" notation. In a somewhat similar fashion, T₄ spaces are often called "normal Hausdorff spaces" by people that wish to avoid the "T" notation. Wikipedia, in particular, wishes to avoid these notations, for the reasons explained in History of the separation axioms[?]. (These conventions can be generalised to other regular 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 won't be discussed here.

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 fully normal if every open cover has an open star refinement[?]. Every fully normal space must also be both normal regular and paracompact. In fact, fully normal spaces actually have more to do with paracompactness than with the usual separation axioms.

X is fully T₄, or fully normal Hausdorff, if it is both T₁ and fully normal. Note that a fully T₄ space must also be T₄.

Sources

• Schechter, Eric; 1997; Handbook of Analysis and its Foundations; http://www.math.vanderbilt.edu/~schectex/ccc/
  • has R_i axioms (among others)
• Willard, Stephen; General Topology; Addison-Wesley
  • has all of the other axioms mentioned in this article, with these definitions
• There are several other good books on general topology, but beware that some use slightly different definitions.