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. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Andrey Nikolayevich Tychonoff (1906–Russian mathematician. ...

The separation axioms are axioms only in the sense that, when defining the notion of topological space, you 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 "Trennung", which means separation. In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. ...

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. In general topology, the separation axioms have had a convoluted history, with many competing meanings for the same term, and many competing terms for the same concept. ...

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.

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. In mathematics, two sets are said to be disjoint if they have no element in common. ... Two or more things are distinct if no two of them are the same thing. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...

Let X be a topological space. Then two subsets A and B of X are separated if each is disjoint from the other's closure. Any two separated sets must be disjoint. In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...

There are several stronger forms of separatedness for sets; they are in order: separated by neighbourhoods; separated by closed neighbourhoods, separated by a function, and separated precisely by a function. These are defined and discussed in the article Separated sets. In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way. ...

Separated sets are distinct from separated spaces, defined below.

We sometimes use the terminology of separated sets to refer to points; in that situation, we're really talking about the singleton set {x} rather than the point x. 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. In mathematics, a singleton is a set with exactly one element. ...

Two points x and y in X are topologically distinguishable if they don't have exactly the same neighbourhoods. If two points are topologically distinguishable, then certainly they are distinct. Furthermore, if the x and y are separated (that is if the singletons {x} and {y} are separated), then they are also topologically distinguishable. ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...

For more on topologically distinguishable points, see Topological distinguishability. In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ...

Definitions of the axioms

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 "T₄" are sometimes interchanged, similarly "regular" and "T₃", etc. Many of the concepts also have several names. In general topology, the separation axioms have had a convoluted history, with many competing meanings for the same term, and many competing terms for the same concept. ...

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 definitions, X is again a topological space, and all functions are supposed to be continuous. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

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. In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ...

X is R₀, or symmetric, if any two topologically distinguishable points in X have disjoint closures. The title given to this article is incorrect due to technical limitations. ...

X is T₁, or accessible or Fréchet, if any two distinct points in X have disjoint closures. Thus, X is T₁ if and only if it is both T₀ and R₀. Although you may say such things as "T₁ 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. The title given to this article is incorrect due to technical limitations. ... This article deals with Fréchet spaces in functional analysis. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...

X is preregular, or R₁, if any two topologically distinguishable points in X are separated by neighbourhoods. An R₁ space must also be R₀. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

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. A Hausdorff space must also be T₁. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

X is T_2½, or Urysohn, if any two distinct points in X are separated by closed neighbourhoods. A T_2½ space must also be Hausdorff. The definition used is this article is in contradiction with the usage of the term elsewhere in Wikipedia. ...

X is completely Hausdorff, or completely T₂, if any two distinct points in X are separated by a function. A completely Hausdorff space must also be T_2½. In topology, completely Hausdorff spaces and Urysohn spaces are types of topological spaces satisfying slightly stronger separation axioms than the more familiar Hausdorff space. ...

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. A regular space must also be R₁. In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ...

X is regular Hausdorff, or T₃, if it is both T₀ and regular. A regular Hausdorff space must also be T_2½. In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ...

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. A completely regular space must also be regular. In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ...

X is Tychonoff, or T_3½, completely T₃, or completely regular Hausdorff, if it is both T₀ and completely regular. A Tychonoff space must also be both regular Hausdorff and completely Hausdorff. In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ...

X is normal if any two disjoint closed subsets of X are separated by neighbourhoods. In fact, in a normal space, any two disjoint closed sets will also be separated by a function; this is Urysohn's lemma. In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... Urysohns lemma in topology states that if X is a normal topological space and A and B are disjoint closed subsets of X, then there exists a continuous function from X into the unit interval [0, 1], f : X → [0, 1], such that f(a) = 0 for all a...

X is normal Hausdorff, or T₄, if it is both T₁ and normal. A normal Hausdorff space must also be both Tychonoff and normal regular. In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ...

X is completely normal if any two separated sets are separated by neighbourhoods. A completely normal space must also be normal. In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ...

X is completely normal Hausdorff, or T₅ or completely T₄, if it is both completely normal and T₁. A T₅ space must also be T₄. In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ...

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. In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ...

X is perfectly normal Hausdorff, or perfectly T₄, if it is both perfectly normal and T₁. A perfectly T₄ space must also be T₅. In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ...

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: In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ...

In this table, you go from the right side to the left side by adding the requirement of T₀, and you go from the left side to the right side by removing that requirement, 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. Download high resolution version (2773x3754, 41 KB) File links The following pages link to this file: Separation axiom ... Look up abbreviation in Wiktionary, the free dictionary Abbreviation (from Latin brevis short) is strictly a shortening, but more particularly, an abbreviation is a letter or group of letters, taken from a word or words, and employed to represent them for the sake of brevity. ...

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". 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 this notation, because it is less likely to be unambiguously understood.) 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. A semiregular space is a topological space whose regular open sets form a base. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases...

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. In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X... In mathematics, in the study in topology of open covers of a topological space X, a star refinement is a particular kind of refinement of an open cover of X. Given x in X and an open cover of X, with index set I, the star of x with respect... In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ...

X is fully T₄, or fully normal Hausdorff, if it is both T₁ and fully normal. A fully T₄ space must also be T₄. In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ...

X is sober if, for every closed set C which 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. In mathematics, particularly in topology, a topological space X is sober if every irreducible closed subset of X is the closure of exactly one singleton of X. An irreducible closed subset of X is defined to be a nonempty closed subset of X which is not the union of two...

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 non-Ri 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.