Kolmogorov space
Encyclopedia
In topology
and related branches of mathematics
, a topological space
X is a T0 space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other. This condition, called the T0 condition, is one of the separation axiom
s. Its intuitive meaning is that the points of X are topologically distinguishable. These spaces are named after Andrey Kolmogorov
.
which contains one of these points and not the other.
Note that topologically distinguishable points are automatically distinct. On the other hand, if the singleton sets {x} and {y} are separated
, then the points x and y must be topologically distinguishable. That is,
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 reverses; points are distinct if and only if
they are distinguishable. This is how the T0 axiom fits in with the rest of the separation axiom
s.
and T1 space
s are T0.
Indeed, when mathematicians in many fields, notably analysis, 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
s f from the real line
R to the complex plane
C such that the Lebesgue integral of |f(x)|2 over the entire real line is finite.
This space should become a normed vector space
by defining the norm ||f|| to be the square root
of that integral. The problem is that this is not really a norm, only a seminorm, because there are functions other than the zero function whose (semi)norms are zero
.
The standard solution is to define L2(R) to be a set of equivalence classes 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.
. 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 naturally homeomorphic.
Categorically, Kolmogorov spaces are a reflective subcategory
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, are exceptions to this rule of thumb.
Even better, many structures 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
, and it has a seminorm, and these define a pseudometric
and a uniform structure 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 T0 since any two functions in L2(R) which are equal almost everywhere
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
.
And it is a Hilbert space that mathematicians (and physicists, in quantum mechanics
) generally want to study. Note that the notation L2(R) usually denotes the Kolmogorov quotient, the set of equivalence classes of square integrable functions which differ on sets of measure zero, rather than simply the vector space of square integrable functions which the notation suggests.
. 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
. 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.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and related branches of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X is a T0 space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other. This condition, called the T0 condition, is one of the separation axiom
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...
s. Its intuitive meaning is that the points of X are topologically distinguishable. These spaces are named after Andrey Kolmogorov
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...
.
Definition
A T0 space is a topological space in which every pair of distinct points is topologically distinguishable. That is, for any two different points x and y there is an open setOpen set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
which contains one of these points and not the other.
Note that topologically distinguishable points are automatically distinct. On the other hand, if the singleton sets {x} and {y} are separated
Separated sets
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....
, 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 reverses; points are distinct if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
they are distinguishable. This is how the T0 axiom fits in with the rest of the separation axiom
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...
s.
Examples and nonexamples
Nearly all topological spaces normally studied in mathematics are T0. In particular, all Hausdorff (T2) spacesHausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
and T1 space
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
s are T0.
Spaces which are not T0
- A set with more than one element, with the trivial topologyTrivial topologyIn topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such a space is sometimes called an indiscrete space, and its topology sometimes called an indiscrete topology...
. 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 topologyProduct topologyIn 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...
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 functionMeasurable functionIn mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
s f from the real lineReal lineIn mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R to the complex planeComplex planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
C such that the Lebesgue integral of |f(x)|2 over the entire real line is finite. Two functions which are equal almost everywhereAlmost everywhereIn measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
are indistinguishable. See also below.
Spaces which are T0 but not T1
- The Zariski topologyZariski topologyIn algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
on Spec(R), the prime spectrum of a commutative ringCommutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
R is always T0 but generally not T1. The non-closed points correspond to prime idealPrime idealIn algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
s which are not maximalMaximal idealIn mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
. They are important to the understanding of schemeScheme (mathematics)In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
s. - The particular point topologyParticular point topologyIn mathematics, the particular point topology is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the particular point topology...
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 spaceSierpinski spaceIn mathematics, the Sierpiński space 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...
which is the particular point topology on the set {0,1}. - The excluded point topologyExcluded point topologyIn mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the excluded point topology on X....
on any set with at least two elements is T0 but not T1. The only closed point is the excluded point. - The Alexandrov topologyAlexandrov topologyIn topology, an Alexandrov space is a topological space 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...
on a partially ordered setPartially ordered setIn mathematics, especially order theory, a partially ordered set 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 relation that indicates that, for certain pairs of elements in the...
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 on a totally ordered set is a related example.
- The overlapping interval topologyOverlapping interval topologyIn mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles.-Definition:Given the closed interval [-1,1] of the real number line, the open sets of the topology are generated from the half-open intervals [-1,b) and In mathematics, the...
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. 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, 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)
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
is meant to be the space of all measurable function
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
s f from the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R to the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
C such that the Lebesgue integral of |f(x)|2 over the entire real line is finite.
This space should become a normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
by defining the norm ||f|| to be the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
of that integral. The problem is that this is not really a norm, only a seminorm, because there are functions other than the zero function whose (semi)norms are zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
.
The standard solution is to define L2(R) to be a set of equivalence classes of functions instead of a set of functions directly.
This constructs a quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
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 relationEquivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
. No matter what topological space X might be to begin with, the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
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 naturally homeomorphic.
Categorically, Kolmogorov spaces are a reflective subcategory
Reflective subcategory
In mathematics, a 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...
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, are exceptions to this rule of thumb.
Even better, many structures 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
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, and it has a seminorm, and these define a pseudometric
Pseudometric
Pseudometric may refer to:* Pseudo-Riemannian manifold* Pseudometric space...
and a uniform structure 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
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
. The space is not T0 since any two functions in L2(R) which are equal almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
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
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
.
And it is a Hilbert space that mathematicians (and physicists, in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
) generally want to study. Note that the notation L2(R) usually denotes the Kolmogorov quotient, the set of equivalence classes of square integrable functions which differ on sets of measure zero, rather than simply the vector space of square integrable functions which 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 HausdorffHausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
. 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 space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
. 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
Pseudometric
Pseudometric may refer to:* Pseudo-Riemannian manifold* Pseudometric space...
. (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.
External links
- History of weak separation axioms (PDF file)