Kolmogorov space

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In
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and related branches of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 points ...
''X'' is a T0 space or Kolmogorov space (named after
Andrey 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 T0 space, all points are
topologically 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 T0 condition, is the weakest of the
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 ...
s. Nearly all topological spaces normally studied in mathematics are T0 spaces. In particular, all T1 spaces, i.e., all spaces in which for every pair of distinct points, each has a neighborhood not containing the other, are T0 spaces. This includes all T2 (or Hausdorff) spaces, i.e., all topological spaces in which distinct points have disjoint neighbourhoods. In another direction, every
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 ...
(which may not be T1) is T0; this includes the underlying topological space of any
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
. Given any topological space one can construct a T0 space by identifying topologically indistinguishable points. T0 spaces that are not T1 spaces are exactly those spaces for which the specialization preorder 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 Applied science, practical discipli ...
, 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 T0 space is a topological space in which every pair of distinct points is
topologically 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 suf ...
that contains one of these points and not the other. More precisely the topological space ''X'' is Kolmogorov or $\mathbf T_0$ if and only if: :If $a,b\in X$ and $a\neq b$, there exists an open set ''O'' such that either $\left(a\in O\right) \wedge \left(b\notin O\right)$ or $\left(a\notin O\right) \wedge \left(b\in O\right)$. 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 T0 space, the second arrow above also reverses; points are distinct
if 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 bicondi ...
they are distinguishable. This is how the T0 axiom fits in with the rest of the
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 ...
s.

# Examples and counter examples

Nearly all topological spaces normally studied in mathematics are T0. In particular, all Hausdorff (T2) spaces, T1 spaces and
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 ...
s are T0.

## Spaces which are not T0

*A set with more than one element, with the
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 R2 where the open sets are the Cartesian product of an open set in R and R itself, i.e., the
product 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-seemin ...
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 di ...
s ''f'' from the
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), 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 ...
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 the ...
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 Lebe ...
$\left\left(\int_ , f\left(x\right), ^2 \,dx\right\right)^ < \infty$. 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 ...

## Spaces which are T0 but not T1

*The
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 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 sp ...
''R'', is always T0 but generally not T1. The non-closed points correspond to
prime 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 with ...
s which are not maximal. They are important to the understanding of
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
s. *The
particular point topology In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collec ...
on any set with at least two elements is T0 but not T1 since the particular point is not closed (its closure is the whole space). An important special case is the
Sierpiński 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 name ...
which is the particular point topology on the set . *The
excluded point topology In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collection :T = \ \cup \ of subsets of ''X'' is then the excluded ...
on any set with at least two elements is T0 but not T1. The only closed point is the excluded point. *The
Alexandrov 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 rest ...
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 (mathematics), set. A poset consists of a set toget ...
is T0 but will not be T1 unless the order is discrete (agrees with equality). Every finite T0 space is of this type. This also includes the particular point and excluded point topologies as special cases. *The
right order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, th ...
on a
totally 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 ( reflexive) ...
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 T0 if and only if the specialization preorder 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 T1 if and only if the order is discrete (i.e. agrees with equality). So a space will be T0 but not T1 if and only if the specialization preorder on ''X'' is a non-discrete partial order.

# Operating with T0 spaces

Examples of topological space typically studied are T0. Indeed, when mathematicians in many fields, notably
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 (38 ...
, naturally run across non-T0 spaces, they usually replace them with T0 spaces, in a manner to be described below. To motivate the ideas involved, consider a well-known example. The space L2(R) is meant to be 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 di ...
s ''f'' from the
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), 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 ...
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 the ...
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 Lebe ...
of , ''f''(''x''), 2 over the entire real line is
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, a verb form that has a subject, usually being inflected or marke ...
. 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 ...
, 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 Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
. The standard solution is to define L2(R) to be a set of
equivalence 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 T0. On the other hand, when ''X'' is fixed but T is allowed to vary within certain boundaries, to force T to be T0 may be inconvenient, since non-T0 topologies are often important special cases. Thus, it can be important to understand both T0 and non-T0 versions of the various conditions that can be placed on a topological space.

# The Kolmogorov quotient

Topological indistinguishability of points is an
equivalence 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 relation ...
. No matter what topological space ''X'' might be to begin with, the quotient space under this equivalence relation is always T0. This quotient space is called the Kolmogorov quotient of ''X'', which we will denote KQ(''X''). Of course, if ''X'' was T0 to begin with, then KQ(''X'') and ''X'' are
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the 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. Although humans are p ...
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 isomorphi ...
. 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'' T0-ness; that is, if ''X'' has such a property, then ''X'' must be T0. Only a few properties, such as being an
indiscrete 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 as ...
s defined on topological spaces can be transferred between ''X'' and KQ(''X''). The result is that, if you have a non-T0 topological space with a certain structure or property, then you can usually form a T0 space with the same structures and properties by taking the Kolmogorov quotient. The example of L2(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
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
, 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 unifor ...
that are compatible with the topology. Also, there are several properties of these structures; for example, the seminorm satisfies the
parallelogram identity In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the s ...
and the uniform structure is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
. The space is not T0 since any two functions in L2(R) that 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 with this topology. When we form the Kolmogorov quotient, the actual L2(R), these structures and properties are preserved. Thus, L2(R) is also a complete seminormed vector space satisfying the parallelogram identity. But we actually get a bit more, since the space is now T0. A seminorm is a norm if and only if the underlying topology is T0, so L2(R) is actually a complete normed vector space satisfying the parallelogram identity—otherwise known as a
Hilbert 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 L2(R) usually denotes the Kolmogorov quotient, the set of
equivalence 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 T0

Although norms were historically defined first, people came up with the definition of seminorm as well, which is a sort of non-T0 version of a norm. In general, it is possible to define non-T0 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
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 mathem ...
. 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 T0-ness from the requirements for a property or structure. It is generally easier to study spaces that are T0, but it may also be easier to allow structures that aren't T0 to get a fuller picture. The T0 requirement can be added or removed arbitrarily using the concept of Kolmogorov quotient.