Separation axiom

In the mathematical field of topology, 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.

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 axiomatisation 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 word "Trennung".

See the Glossary at the end of this article for a definitive list of axioms and links to those axioms with their own articles on Wikipedia.

Definitions of the main axioms

Please read Modern Terminology below before bandying about these terms!

See also Other Separation Axioms below.

As originally formulated, the separation axioms are:

T₁ axiom

Given any distinct points x and y in the topological space, there are neighbourhoods U of x and V of y such that x does not belong to V and y does not belong to U.

T₂ axiom

Given any distinct points x and y in the topological space, there are neighbourhoods U of x and V of y such that U and V are disjoint.

T₃ axiom

Given any point x and closed set F in the topological space such that x does not belong to F, there are neighbourhoods U of x and V of F such that U and V are disjoint.

T₄ axiom

Given any disjoint closed sets E and F in the topological space, there are neighbourhoods U of E and V of F such that U and V are disjoint.

Later, some more were added:

T₀ axiom

Given any distinct points x and y in the topological space, there are neighbourhoods U of x and V of y such that x does not belong to V or y does not belong to U.

T_2½ axiom

Given any distinct points x and y in the topological space, there are closed neighbourhoods U of x and V of y such that U and V are disjoint.

T_3½ axiom

Given any point x and closed set F in the topological space X such that x does not belong to F, there is a continuous function f from X to the real line R such that f(x) = 0 and f(F) = {1}.

T₅ axiom

Every subspace of the topological space satisfies the T₄ axiom.

Elementary results

To explain why the modern terminology in the next section is what it is, one needs to understand this section, or at least its last paragraph. Here X is a topological space.

First consider the modification from the T₄ axiom to the T₅ axiom, asking for the axiom to be satisfied by every subspace. Why isn't this modification made to any of the other axioms? Because it does nothing to them:

• If X satisfies any separation axiom other than T₄, then so does every subspace of X.

Now consider the modification from the T₃ axiom to the T_3½ axiom, in which asking the neighbourhoods to be disjoint is replaced with asking for a continuous function that witnesses this fact. Why isn't this modification made to any of the other applicable axioms? When made to the T₂ axiom, this modification yields the Urysohn axiom (see Other Separation Axioms), but it does nothing to the T₄ and T₅ axioms:

• If X satisfies the T₄ axiom and E and F are disjoint closed sets in X, then there is a continuous function f from X to the real line R such that f(E) = {0} and f(F) = {1}. (This is the Urysohn lemma and is the only statement here that is difficult to prove.)
• If X satisfies the T₅ axiom, then every subspace of X has the property in the previous bullet point.

Now consider the modification from the T₂ axiom to the T_2½ axiom, in which the neighbourhoods are required to be closed. Why isn't this modification made to any of the other axioms? It either does nothing to the axiom or strengthens it to T₂:

• If X satisfies any separation axiom T_i for i > 2½, then it satisfies the same axiom with "neighbourhoods" replaced by "closed neighbourhoods".
• If X satisfies the T₂ axiom, then it satisfies T₀ and T₁ with "neighbourhoods" replaced by "closed neighbourhoods".
• If X satisfies the T₀ or T₁ axioms with "neighbourhoods" replaced by "closed neighbourhoods", then it satisfies the T₂ axiom.

Finally, notice that:

• X satisfies the T₀ axiom iff, given any distinct points x and y in X, x does not belong to the closure of {y} or y does not belong to the closure of {x}.
• X satisfies the T₁ axiom iff any point {x} is closed.

Thus to end this section, we get these implications:

• If X satisfies T₅, then it satisfies T₄.
• If X satisfies T₄ and T₃, then it satisfies T_3½.
• If X satisfies T₄ and T₁, then it satisfies T_3½.
• If X satisfies T_3½, then it satisfies T₃.
• If X satisfies T₃ and T₀, then it satisfies T_2½.
• If X satisfies T_2½, then it satisfies T₂.
• If X satisfies T₂, then it satisfies T₁.
• If X satisfies T₁, then it satisfies T₀.

Modern terminology

It's clear from the last list of bullet points that a really nice list of properties is the following, in which each property is strictly stronger than the previous one:

• X satisfies T₀;
• X satisfies T₁;
• X satisfies T₂;
• X satisfies T_2½;
• X satisfies T₃ and T₀;
• X satisfies T_3½ and T₀;
• X satisfies T₄ and T₁;
• X satisfies T₅ and T₁.

To discuss these, terminology like the following has developed:

Axiom	Common Noun	Person	Added Axiom
T0		Kolmogorov
T1		Fréchet
T2	separated	Hausdorff
T2½	completely separated	Urysohn
T3	regular		T0
T3½	completely regular	Tychonoff	T0
T4	normal		T1
T5	completely normal		T1

The axioms in the Added Axiom column are those that must be added to the axioms in the Axiom column to create the "nice" list of properties above.

With the exception of "separated", the terms in the Common Noun column are very commonly used. With the exception of Hausdorff, the mathematicians in the Person column find their names used more rarely.

Leaving the Person column aside for the moment, in the rows T₃ and higher, there are two concepts that require naming: satisfying the axiom in the Axiom column, and satisfying both the axiom in the Axiom column and the axiom in the Added Axiom column. The most natural solution, given the terms in the table, is to say "T_i" when the T_i axiom alone is satisfied and then use the term in the Common Noun column when the axiom in the Added Axiom column is also satisfied, and this is exactly what was originally done. Such usage can be found, for example, in Steen & Seebach (see Sources below).

However, this means that the most "nice" properties don't have the simplest names. So it is most common nowadays to say that a space "is" T_i or is a "T_i space" (as opposed to merely "satisfies" the T_i axiom) for some i when it satisfies both the axiom T_i and the corresponding axiom (if any) in the Added Axiom column. Then the term in the Common Noun column can be used for a space satisfying only the axiom T_i. Thus, the meanings have been switched. The modern usage can be found, for example, in Willard (see Sources). It is also the usage adopted in Wikipedia.

As for the Person column, with the exception of "separated" and "Hausdorff", which are always interchangeable, the terms in the Common Noun column usually don't mean the same thing as the name of the mathematicians in the Person column. Unfortunately, again there is no agreement on what the difference is. Originally, "completely separated" referred to the T_2½ axiom, while "Urysohn" refered to the Urysohn axiom listed in Other Separation Axioms below. But the usage is usually reversed nowadays. On the other hand, "Tychonoff" originally referred to only the T_3½ axiom, while today it usually implies the T₀ axiom as well. Thus, the meaning of "completely regular" and "Tychonoff" have also been switched.

To make matters even more confusing, just because an author has chosen the more modern interpretation of (say) what "Tychonoff" means doesn't mean that they have chosen the more modern interpretation of what (say) "regular" means. There are probably five or six systems in use.

But the bottom line is this: When reading a book, paper, or web site about topology, always make sure that you know which definitions the author is using!!!

(A final note: The adjective "completely" can be placed before "Hausdorff", "T₂", "T₃", and "T₄" as well as before "separated", "regular", and "normal". So "completely T₃" means the same as "T_3½", whatever that may be. This just gives each system some alternative words but fortunately doesn't lead to an explosion of new systems of terminology. A glossary of all possible terms appears at the end of this article.)

Other separation axioms

There are a few other related axioms that don't fit into this numbering scheme. These are:

Perfect T₄ axiom

Given any disjoint closed sets A and B, there is a continuous function f from X to the real line R such that f ^-1({0}) = A and f ^-1({1}) = B.

Originally, a topological space that satisfied the perfect T₄ axiom was called "perfectly T₄", but it's now usually called "perfectly normal". You get the other term (whichever it may be) by adding the requirement that the space satisfy T₁.

Any space that satisfies the perfect T₄ axiom must also satisfy the T₅ axiom. If you try to analogously define a "perfect T₃" axiom, you actually get back the T_3½ axiom. Similarly, if you try to analogously define a "perfect T₂" axiom, you get the Urysohn axiom from below in this section.

Urysohn axiom

Given any distinct points x and y in X, there is a continuous function f from X to the real line R such that f(x) = 0 and f(y) = 1.

As mentioned in the section Modern Terminology above, spaces satisfying the Urysohn axiom were originally called "Urysohn" spaces, but they are now more often called "completely separated" (or "completely Hausdorff", or "completely T₂"), with the term "Urysohn space" used more generally for any space that satisfies T_2½.

Any topological space satisfying both T_3½ and T₀ must satisfy the Urysohn axiom, and any space that satisfies the Urysohn axiom must satisfy the T_2½ axiom.

Semiregularity axiom

First note that if the topological space X satisfies T₃, then the regular open sets form a base for the open sets on X. This may be the origin of the association of T₃ with the term "regular". In any case, X is semiregular iff the regular open sets form a base. But the term "semiregular" was originally reserved for a space that also satisfies T₂, adding another level to the terminological confusion.

R₁ axiom

Given any points x and y in X, either there are neighbourhoods U of x and V of y such that U and V are disjoint or x belongs to the closure of {y} and y belongs to the closure of {x}.

Thus a space satisfies the T₂ axiom iff it satisfies both T₀ and R₁ axioms. Any space satisfying the T₃ axiom must satisfy R₁. Essentially, R₁ is T₂ without the T₀.

A topological space is preregular iff it satisfies R₁.

R₀ axiom

Given any points x and y in X, x belongs to the closure of {y} if and only if y belongs to the closure of {x}.

Thus a space satisfies the T₁ axiom iff it satisfies both the T₀ and R₀ axioms. Any space satisfying R₁ must satisfy R₀, and any space satisfying the T₄ and R₀ axioms must satisfy the T_3½ axiom.

A topological space is symmetric iff it satisfies the R₀ axiom.

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These last two properties fit into an alternative sequence of "nice" properties, as in the following table:

Usual nice property	Alternative nice property
T0 axiom	No requirement
T1 axiom	R0 axiom
T2 axiom	R1 axiom
T3 and T0 axioms	T3 axiom
T3½ and T0 axioms	T3½ axiom
T4 and T1 axioms	T4 and R0 axioms

In this table, each row is stronger than the one above it, and you get the properties in the left column by adding the requirement T₀ to the properties in the right column.

Diagram

A diagram of the relationships between the separation axioms is under construction.

Glossary

Here, in alphabetical order, are all the terms that are used to describe a topological space with a certain separation property, to the knowledge of the authors of this article so far. The definitions are given in terms of which axioms they satisfy. Many terms have two definitions in use, an older definition and a more modern definition; we give both for your reference, but Wikipedia always follows the more modern definition. The modern definitions of some of these terms can also be found in the Topology Glossary. Links are provided to those types of spaces that have articles of their own in Wikipedia.

Type of space	Axioms required in:
	Original definition	Modern definition (= Wikipedia usage)
completely Hausdorff	T2½	Urysohn axiom
completely normal	T5, T1	T5
completely regular	T3½, T0	T3½
completely separated	T2½	Urysohn axiom
completely T2 space	T2½	Urysohn axiom
completely T3 space	T3½	T3½, T0
completely T4 space	T5	T5, T1
Fréchet	T1 (or else completely unrelated notion of Fréchet space from functional analysis)
Hausdorff	T2
Kolmogorov	T0
normal	T4, T1	T4
perfectly normal	perfect T4, T1	perfect T4
perfectly T4 space	perfect T4	T4, T1
preregular	R1
R0 space	R0
R1 space	R1
regular	T3, T0	T3
semiregular	semiregularity axiom, T2	semiregularity axiom
symmetric	R0
separated	T2
T0 space	T0
T1 space	T1
T2 space	T2
T2½ space	T2½
T3 space	T3	T3, T0
T3½ space	T3½	T3½, T0
T4 space	T4	T4, T1
T5 space	T5	T5, T1
Tychonoff	T3½	T3½, T0
Urysohn	Urysohn axiom	T2½

Sources

• Schechter, Eric; 1997; Handbook of Analysis and its Foundations; Electronic Edition; Academic Press and Aztec Corporation: Waltham, MA (1998)
  • has R_i axioms
• Steen, Lynn Arthur, & Seebach, J. Arthur, Jr.; 1978; Counterexamples in Topology; Second Edition; Dover: New York (1995)
  • standard reference with older terminology
• Willard, Stephen; General Topology; Addison-Wesley
  • standard reference with newer terminology