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Intopology and related branches ofmathematics, atopological spaceX is aT₀ space orKolmogorov space (named afterAndrey Kolmogorov) if for every pair of distinct points ofX, at least one of them has aneighborhood not containing the other.^[1] In a T₀ space, all points aretopologically distinguishable.

This condition, called theT₀ condition, is the weakest of theseparation axioms. Nearly all topological spaces normally studied in mathematics are T₀ spaces. In particular, allT₁ 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 allT₂ (or Hausdorff) spaces, i.e., all topological spaces in which distinct points have disjoint neighbourhoods. In another direction, everysober space (which may not be T₁) is T₀; this includes the underlying topological space of anyscheme. 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 thespecialization preorder is a nontrivialpartial order. Such spaces naturally occur incomputer science, specifically indenotational semantics.

AT₀ space is a topological space in which every pair of distinct points istopologically distinguishable. That is, for any two different pointsx andy there is anopen set that contains one of these points and not the other. More precisely the topological spaceX is Kolmogorov or${\displaystyle {\mathbf{T}}_0}$ if and only if:^[1]

If${\displaystyle a,b\in X}$ and${\displaystyle a\ne b}$, there exists an open setO such that either${\displaystyle (a\in O)\wedge (b\notin O)}$ or${\displaystyle (a\notin O)\wedge (b\in O)}$.

Note that topologically distinguishable points are automatically distinct. On the other hand, if thesingleton sets {x} and {y} areseparated then the pointsx andy 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 distinctif and only if they are distinguishable. This is how the T₀ axiom fits in with the rest of theseparation axioms.

Nearly all topological spaces normally studied in mathematics are T₀. In particular, allHausdorff (T₂) spaces,T₁ spaces andsober spaces are T₀.

• A set with more than one element, with thetrivial topology. No points are distinguishable.
• The setR² where the open sets are the Cartesian product of an open set inR andR itself, i.e., theproduct topology ofR with the usual topology andR with the trivial topology; points (a,b) and (a,c) are not distinguishable.
• The space of allmeasurable functionsf from thereal lineR to thecomplex planeC such that theLebesgue integral. Two functions which are equalalmost everywhere are indistinguishable. See also below.

• TheZariski topology on Spec(R), theprime spectrum of acommutative ringR, is always T₀ but generally not T₁. The non-closed points correspond toprime ideals which are notmaximal. They are important to the understanding ofschemes.
• Theparticular 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 theSierpiński space which is the particular point topology on the set {0,1}.
• Theexcluded point topology on any set with at least two elements is T₀ but not T₁. The only closed point is the excluded point.
• TheAlexandrov topology on apartially ordered set 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.
• Theright order topology on atotally ordered set is a related example.
• Theoverlapping interval topology is similar to the particular point topology since every non-empty open set includes 0.
• Quite generally, a topological spaceX will be T₀ if and only if thespecialization preorder onX is apartial 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 onX is a non-discrete partial order.

Commonly studied topological spaces are all T₀.Indeed, when mathematicians in many fields, notablyanalysis, 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 spaceL²(R) is meant to be the space of allmeasurable functionsf from thereal lineR to thecomplex planeC such that theLebesgue integral of |f(x)|² over the entire real line isfinite.This space should become anormed vector space by defining the norm ||f|| to be thesquare root of that integral. The problem is that this is not really a norm, only aseminorm, because there are functions other than thezero function whose (semi)norms arezero.The standard solution is to define L²(R) to be a set ofequivalence classes of functions instead of a set of functions directly.This constructs aquotient 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 topologyT on a setX, it is helpful if that topology is T₀. On the other hand, whenX is fixed butT is allowed to vary within certain boundaries, to forceT 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.

Topological indistinguishability of points is anequivalence relation. No matter what topological spaceX might be to begin with, thequotient space under this equivalence relation is always T₀. This quotient space is called theKolmogorov quotient ofX, which we will denote KQ(X). Of course, ifX was T₀ to begin with, then KQ(X) andX arenaturallyhomeomorphic.Categorically, Kolmogorov spaces are areflective subcategory of topological spaces, and the Kolmogorov quotient is the reflector.

Topological spacesX andY areKolmogorov equivalent when their Kolmogorov quotients are homeomorphic. Many properties of topological spaces are preserved by this equivalence; that is, ifX andY are Kolmogorov equivalent, thenX has such a property if and only ifY does.On the other hand, most of theother properties of topological spacesimply T₀-ness; that is, ifX has such a property, thenX must be T₀.Only a few properties, such as being anindiscrete space, are exceptions to this rule of thumb.Even better, manystructures defined on topological spaces can be transferred betweenX 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 avector space, and it has a seminorm, and these define apseudometric and auniform structure that are compatible with the topology.Also, there are several properties of these structures; for example, the seminorm satisfies theparallelogram identity and the uniform structure iscomplete. The space is not T₀ since any two functions in L²(R) that are equalalmost 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 aHilbert space.And it is a Hilbert space that mathematicians (andphysicists, inquantum mechanics) generally want to study. Note that the notation L²(R) usually denotes the Kolmogorov quotient, the set ofequivalence classes 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.

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 beingHausdorff. One can then define another property of topological spaces by defining the spaceX 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 spaceX is calledpreregular. (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 ametric. We can define a new structure on topological spaces by letting an example of the structure onX be simply a metric on KQ(X). This is a sensible structure onX; it is apseudometric. (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.

• Sober space

• 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

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