In _{0} space or Kolmogorov space (named after _{0} space, all points are _{0} condition, is the weakest of the _{0} spaces. In particular, all T_{1} spaces, i.e., all spaces in which for every pair of distinct points, each has a neighborhood not containing the other, are T_{0} spaces. This includes all T_{2} (or Hausdorff) spaces, i.e., all topological spaces in which distinct points have disjoint neighbourhoods. In another direction, every _{1}) is T_{0}; this includes the underlying topological space of any scheme. Given any topological space one can construct a T_{0} space by identifying topologically indistinguishable points.
T_{0} spaces that are not T_{1} spaces are exactly those spaces for which the

_{0} space is a topological space in which every pair of distinct points is _{0} space, the second arrow above also reverses; points are distinct _{0} axiom fits in with the rest of the

_{0}. In particular, all Hausdorff (T_{2}) spaces, T_{1} spaces and _{0}.

^{2} where the open sets are the Cartesian product of an open set in R and R itself, i.e., the

_{0} but generally not T_{1}. The non-closed points correspond to _{0} but not T_{1} since the particular point is not closed (its closure is the whole space). An important special case is the _{0} but not T_{1}. The only closed point is the excluded point.
*The _{0} but will not be T_{1} unless the order is discrete (agrees with equality). Every finite T_{0} space is of this type. This also includes the particular point and excluded point topologies as special cases.
*The right order topology on a _{0} if and only if the _{1} if and only if the order is discrete (i.e. agrees with equality). So a space will be T_{0} but not T_{1} if and only if the specialization preorder on ''X'' is a non-discrete partial order.

_{0}.
Indeed, when mathematicians in many fields, notably _{0} spaces, they usually replace them with T_{0} spaces, in a manner to be described below. To motivate the ideas involved, consider a well-known example. The space L^{2}(R) is meant to be the space of all ^{2} over the entire real line is ^{2}(R) to be a set of _{0}. On the other hand, when ''X'' is fixed but T is allowed to vary within certain boundaries, to force T to be T_{0} may be inconvenient, since non-T_{0} topologies are often important special cases. Thus, it can be important to understand both T_{0} and non-T_{0} versions of the various conditions that can be placed on a topological space.

_{0}. This quotient space is called the Kolmogorov quotient of ''X'', which we will denote KQ(''X''). Of course, if ''X'' was T_{0} to begin with, then KQ(''X'') and ''X'' are _{0}-ness; that is, if ''X'' has such a property, then ''X'' must be T_{0}.
Only a few properties, such as being an _{0} topological space with a certain structure or property, then you can usually form a T_{0} space with the same structures and properties by taking the Kolmogorov quotient.
The example of L^{2}(R) displays these features.
From the point of view of topology, the seminormed vector space that we started with has a lot of extra structure; for example, it is a _{0} since any two functions in L^{2}(R) that are equal ^{2}(R), these structures and properties are preserved.
Thus, L^{2}(R) is also a complete seminormed vector space satisfying the parallelogram identity.
But we actually get a bit more, since the space is now T_{0}.
A seminorm is a norm if and only if the underlying topology is T_{0}, so L^{2}(R) is actually a complete normed vector space satisfying the parallelogram identity—otherwise known as a ^{2}(R) usually denotes the Kolmogorov quotient, the set of

_{0} version of a norm. In general, it is possible to define non-T_{0} versions of both properties and structures of topological spaces. First, consider a property of topological spaces, such as being Hausdorff. One can then define another property of topological spaces by defining the space ''X'' to satisfy the property if and only if the Kolmogorov quotient KQ(''X'') is Hausdorff. This is a sensible, albeit less famous, property; in this case, such a space ''X'' is called '' preregular''. (There even turns out to be a more direct definition of preregularity). Now consider a structure that can be placed on topological spaces, such as a _{0}-ness from the requirements for a property or structure. It is generally easier to study spaces that are T_{0}, but it may also be easier to allow structures that aren't T_{0} to get a fuller picture. The T_{0} requirement can be added or removed arbitrarily using the concept of Kolmogorov quotient.

topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

and related branches of mathematics, a topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

''X'' is a TAndrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...

) if for every pair of distinct points of ''X'', at least one of them has a neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

not containing the other. In a Ttopologically distinguishable
In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...

.
This condition, called the Tseparation 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 ...

s. Nearly all topological spaces normally studied in mathematics are Tsober space In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible closed subset of ''X'' is the closure of exactly one point of ''X'': that is, every irreducible closed subset has a unique generic point.
Definitio ...

(which may not be Tspecialization preorder In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space. For most spaces that are considered in practice, namely for all those that satisfy the ...

is a nontrivial partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

. Such spaces naturally occur in computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...

, specifically in denotational semantics
In computer science, denotational semantics (initially known as mathematical semantics or Scott–Strachey semantics) is an approach of formalizing the meanings of programming languages by constructing mathematical objects (called ''denotations'' ...

.
Definition

A Ttopologically distinguishable
In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...

. That is, for any two different points ''x'' and ''y'' there is an open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...

that contains one of these points and not the other. More precisely the topological space ''X'' is Kolmogorov or $\backslash mathbf\; T\_0$ if and only if:
:If $a,b\backslash in\; X$ and $a\backslash neq\; b$, there exists an open set ''O'' such that either $(a\backslash in\; O)\; \backslash wedge\; (b\backslash notin\; O)$ or $(a\backslash notin\; O)\; \backslash wedge\; (b\backslash in\; O)$.
Note that topologically distinguishable points are automatically distinct. On the other hand, if the singleton set
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermelo–Fraenkel set theory, the ...

s and are separated then the points ''x'' and ''y'' must be topologically distinguishable. That is,
:''separated'' ⇒ ''topologically distinguishable'' ⇒ ''distinct''
The property of being topologically distinguishable is, in general, stronger than being distinct but weaker than being separated. In a Tif and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...

they are distinguishable. This is how the Tseparation 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 ...

s.
Examples and counter examples

Nearly all topological spaces normally studied in mathematics are Tsober space In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible closed subset of ''X'' is the closure of exactly one point of ''X'': that is, every irreducible closed subset has a unique generic point.
Definitio ...

s are T Spaces which are not T_{0}

trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...

. No points are distinguishable.
*The set Rproduct topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...

of R with the usual topology and R with the trivial topology; points (''a'',''b'') and (''a'',''c'') are not distinguishable.
*The space of all measurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in d ...

s ''f'' from the real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...

R to the complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...

C such that the Lebesgue integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...

$\backslash left(\backslash int\_\; ,\; f(x),\; ^2\; \backslash ,dx\backslash right)^\; <\; \backslash infty$. Two functions which are equal almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...

are indistinguishable. See also below.
Spaces which are T_{0} but not T_{1}

Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...

on Spec(''R''), the prime spectrum
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...

of a commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...

''R'', is always Tprime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...

s which are not maximal. They are important to the understanding of schemes.
*The particular point topology on any set with at least two elements is TSierpiński space
In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed.
It is the smallest example of a topological space which is neither trivial nor discrete. It is named ...

which is the particular point topology on the set .
*The excluded point topology on any set with at least two elements is TAlexandrov topology In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any ''finite'' family of open sets is open; in Alexandrov topologies the finite restr ...

on a partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

is Ttotally ordered set
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexiv ...

is a related example.
*The overlapping interval topology
In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles.
Definition
Given the closed interval 1,1/math> of the real number line, the open sets of the topology are generated from ...

is similar to the particular point topology since every open set includes 0.
*Quite generally, a topological space ''X'' will be Tspecialization preorder In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space. For most spaces that are considered in practice, namely for all those that satisfy the ...

on ''X'' is a partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

. However, ''X'' will be T Operating with T_{0} spaces

analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...

, naturally run across non-Tmeasurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in d ...

s ''f'' from the real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...

R to the complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...

C such that the Lebesgue integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...

of , ''f''(''x''), finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb
Traditionally, a finite verb (from la, fīnītus, past particip ...

.
This space should become a normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...

by defining the norm , , ''f'', , to be the square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...

of that integral. The problem is that this is not really a norm, only a seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk an ...

, because there are functions other than the zero function
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...

whose (semi)norms are zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...

.
The standard solution is to define Lequivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...

es of functions instead of a set of functions directly.
This constructs a quotient space of the original seminormed vector space, and this quotient is a normed vector space. It inherits several convenient properties from the seminormed space; see below.
In general, when dealing with a fixed topology T on a set ''X'', it is helpful if that topology is TThe Kolmogorov quotient

Topological indistinguishability of points is anequivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...

. No matter what topological space ''X'' might be to begin with, the quotient space under this equivalence relation is always Tnatural
Nature, in the broadest sense, is the physics, physical world or universe. "Nature" can refer to the phenomenon, phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. ...

ly homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorp ...

.
Categorically, Kolmogorov spaces are a reflective subcategory
In mathematics, a full subcategory ''A'' of a category ''B'' is said to be reflective in ''B'' when the inclusion functor from ''A'' to ''B'' has a left adjoint. This adjoint is sometimes called a ''reflector'', or ''localization''. Dually, ''A' ...

of topological spaces, and the Kolmogorov quotient is the reflector.
Topological spaces ''X'' and ''Y'' are Kolmogorov equivalent when their Kolmogorov quotients are homeomorphic. Many properties of topological spaces are preserved by this equivalence; that is, if ''X'' and ''Y'' are Kolmogorov equivalent, then ''X'' has such a property if and only if ''Y'' does.
On the other hand, most of the ''other'' properties of topological spaces ''imply'' Tindiscrete space In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...

, are exceptions to this rule of thumb.
Even better, many structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such ...

s defined on topological spaces can be transferred between ''X'' and KQ(''X'').
The result is that, if you have a non-Tvector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...

, and it has a seminorm, and these define a pseudometric and a uniform structure
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unif ...

that are compatible with the topology.
Also, there are several properties of these structures; for example, the seminorm satisfies the parallelogram identity and the uniform structure is complete. The space is not Talmost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...

are indistinguishable with this topology.
When we form the Kolmogorov quotient, the actual LHilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

.
And it is a Hilbert space that mathematicians (and physicists
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate caus ...

, in quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

) generally want to study. Note that the notation Lequivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...

es of square integrable functions that differ on sets of measure zero, rather than simply the vector space of square integrable functions that the notation suggests.
Removing T_{0}

metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...

. We can define a new structure on topological spaces by letting an example of the structure on ''X'' be simply a metric on KQ(''X''). This is a sensible structure on ''X''; it is a pseudometric. (Again, there is a more direct definition of pseudometric.)
In this way, there is a natural way to remove TSee also

*Sober space In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible closed subset of ''X'' is the closure of exactly one point of ''X'': that is, every irreducible closed subset has a unique generic point.
Definitio ...

References

*Lynn Arthur Steen and J. Arthur Seebach, Jr., ''Counterexamples in Topology''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. {{ISBN, 0-486-68735-X (Dover edition). Separation axioms Properties of topological spaces