In topology and related branches of mathematics, the T₀ spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class of well-behaved topological spaces. The T₀ condition is one of the separation axioms. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... 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. ... The title given to this article is incorrect due to technical limitations. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... The definition used is this article is in contradiction with the usage of the term elsewhere in Wikipedia. ... In topology, completely Hausdorff spaces and Urysohn spaces are types of topological spaces satisfying slightly stronger separation axioms than the more familiar Hausdorff space. ... In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Andrey Nikolaevich Kolmogorov (Russian: Андре́й Никола́евич Колмого́ров) (April 25, 1903 - October 20, 1987) was a Soviet mathematician who made major advances in different academic fields (among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity). ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... 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. ...

Topological distinguishability

To define T₀ spaces, we first define the concept of topological distinguishability. If X is a topological space and x and y are points in X, then x and y are topologically indistinguishable if and only if any of the following equivalent conditions hold: Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ...

• they have exactly the same neighbourhoods; i.e. for all open sets U, we have x ∈ U if and only if y ∈ U,
• x belongs to the closure of {y} and y belongs to the closure of {x},
• {x} and {y} have the same closures.

Otherwise, x and y are said to be topologically distinguishable. This is a glossary of some terms used in the branch of mathematics known as topology. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... ↔ ⇔ ≡ logical symbols representing iff. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...

In an indiscrete space, for example, any two points are topologically indistinguishable. In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. ...

Note that the closure of a point x contains all points indistinguishable from x. It may contain other points as well.

Definition

The definition of a T₀ space is now simple; X is T₀ if and only if every pair of distinct points is topologically distinguishable. Two or more things are distinct if no two of them are the same thing. ...

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, In mathematics, a singleton is a set with exactly one element. ... 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. ...

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 they are distinguishable. This is how the T₀ axiom fits in with the rest of the separation axioms. ↔ ⇔ ≡ logical symbols representing iff. ... 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. ...

This definition may also be formulated as follows: X is a T₀ space if and only if for any two distinct points in X there exists an open subset of X which contains one of the points but not the other. This characterisation should be contrasted with an analogous characterisation of T₁ spaces, where one can specify beforehand which points will belong to the open set. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... The title given to this article is incorrect due to technical limitations. ...

Examples of topological spaces which are not T₀

• A set with more than one element, with the trivial 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 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 functions 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. Two functions which are equal almost everywhere are indistinguishable. See also below.

In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ... In mathematics, the real line is simply the set of real numbers. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...

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) is meant to be the space of all measurable functions 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. Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ... In mathematics, the real line is simply the set of real numbers. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... 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. ... In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of length. A space with such a seminorm is then known as a seminormed space. ... Zero redirects here. ... For other senses of this word, see zero or 0. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ...

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. 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 homeomorphic. In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... This word should not be confused with homomorphism. ...

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. In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. ... In mathematics, a structure on a set is some additional mathematical objects that, loosely speaking, attach to the set, making it easier to visualize or work with. ...

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. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ... In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ... The parallelogram law in elementary geometry In elementary geometry, the parallelogram law states that the sum of the squares of the lengths of the fours sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... Many famous physicists of the 20th and 21st century are found on the list of recipients of the Nobel Prize in physics. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

Removing T₀

You may notice that, 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. 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 topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ...

In this way, there is a natural way to remove T₀-ness from the requirements for a property or structure. It's 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

• The Legacy of Andrei Nikolaevich Kolmogorov Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A.N. Kolmogorov. A.N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A.N. Kolmogorov.