In
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
and related branches of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X'' is a T
0 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 Soviet ...
) if for every pair of distinct points of ''X'', at least one of them has a
neighborhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
not containing the other.
In a T
0 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 T
0 condition, is the weakest of the
separation axioms. Nearly all topological spaces normally studied in mathematics are T
0 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 T
0 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 space, irreducible closed subset of ''X'' is the closure (topology), closure of exactly one point of ''X'': that is, every nonempty irreducible close ...
(which may not be T
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
specialization preorder is a nontrivial
partial order
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable ...
. Such spaces naturally occur in
computer science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
, 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 T
0 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, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
that contains one of these points and not the other. More precisely the topological space ''X'' is Kolmogorov or
if and only if:
:If
and
, there exists an open set ''O'' such that either
or
.
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 a ...
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 T
0 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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
they are distinguishable. This is how the T
0 axiom fits in with the rest of the
separation axioms.
Examples and counter examples
Nearly all topological spaces normally studied in mathematics are T
0. 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 space, irreducible closed subset of ''X'' is the closure (topology), closure of exactly one point of ''X'': that is, every nonempty irreducible close ...
s are T
0.
Spaces that 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 conseque ...
. No points are distinguishable.
*The set R
2 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 ...
s ''f'' from the
real line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
R to the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
C such that the
Lebesgue integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
. 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 that are T0 but not T1
*The
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...
on Spec(''R''), the
prime spectrum
In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative 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 ...
of a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), 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 prope ...
''R'', is always T
0 but generally not T
1. The non-closed points correspond to
prime ideal
In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
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 T
0 but not T
1 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 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 after Wacław Sierpiński.
The ...
which is the particular point topology on the set .
*The
excluded point topology In mathematics, the excluded point topology is a topological space, topology where exclusion of a particular point defines open set, openness. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collection
:T = \ \cup \
of subsets of ...
on any set with at least two elements is T
0 but not T
1. The only closed point is the excluded point.
*The
Alexandrov topology
In general topology, an Alexandrov topology is a topology in which the intersection of an ''arbitrary'' family of open sets is open (while the definition of a topology only requires this for a ''finite'' family). Equivalently, an Alexandrov top ...
on a
partially ordered set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
is T
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
totally ordered set
In mathematics, a total order 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 ( ref ...
is a related example.
*The
overlapping interval topology is similar to the particular point topology since every non-empty open set includes 0.
*Quite generally, a topological space ''X'' will be T
0 if and only if the
specialization preorder on ''X'' is a
partial order
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable ...
. However, ''X'' will be T
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.
Operating with T0 spaces
Commonly studied topological spaces are all T
0.
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-T
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
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 ...
s ''f'' from the
real line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
R to the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
C such that the
Lebesgue integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...
of , ''f''(''x''),
2 over the entire real line is
finite.
This space should become a
normed vector space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898.
The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war ...
by defining the norm , , ''f'', , to be the
square root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
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 like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
, because there are functions other than the
zero function
0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
whose (semi)norms are
zero
0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
.
The standard solution is to define L
2(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 ...
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 T
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.
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. A simpler example is equ ...
. No matter what topological space ''X'' might be to begin with, the
quotient space under this equivalence relation is always T
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
natural
Nature is an inherent character or constitution, particularly of the ecosphere or the universe as a whole. In this general sense nature refers to the laws, elements and phenomena of the physical world, including life. Although humans are part ...
ly
homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
.
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'' T
0-ness; that is, if ''X'' has such a property, then ''X'' must be T
0.
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-T
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
vector 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'', can be added together and multiplied ("scaled") by numbers called sc ...
, 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 topological space, set with additional mathematical structure, structure that is used to define ''uniform property, uniform properties'', such as complete space, completeness, uniform con ...
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. The space is not T
0 since any two functions in L
2(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 L
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
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
.
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 cau ...
, in
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
) generally want to study. Note that the notation L
2(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 ...
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-T
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
metric
Metric or metrical may refer to:
Measuring
* 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
...
. 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 T
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.
See also
*
Sober space
In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible space, irreducible closed subset of ''X'' is the closure (topology), closure of exactly one point of ''X'': that is, every nonempty irreducible close ...
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