Kolmogorov space

2024-08-08T02:06:45Z

2400:A845:46CC:0:0:CC64:548A:42: /* Spaces which are T0 but not T1 */

{{Short description|Concept in topology}}
{{More citations needed|date=April 2022}}
{{Separation axioms}}
In [[topology]] and related branches of [[mathematics]], a [[topological space]] ''X'' is a '''T0 space''' or '''Kolmogorov space''' (named after [[Andrey Kolmogorov]]) if for every pair of distinct points of ''X'', at least one of them has a [[Neighbourhood (mathematics)|neighborhood]] not containing the other.<ref name=":0">{{Cite journal |last=Karno |first=Zbigniew |date=1994 |title=On Kolmogorov Topological Spaces |url=https://mizar.uwb.edu.pl/JFM/pdf/tsp_1.pdf |journal=Journal of Formalized Mathematics |publication-date=2003 |volume=6}}</ref> In a T0 space, all points are [[topologically distinguishable]].

This condition, called the '''T0 condition''', is the weakest of the [[separation axiom]]s. Nearly all topological spaces normally studied in mathematics are T0 spaces. In particular, all [[T1 space|T1 space]]s, 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 [[Hausdorff space|T2 (or Hausdorff) spaces]], i.e., all topological spaces in which distinct points have disjoint neighbourhoods. In another direction, every [[sober space]] (which may not be T1) is T0; this includes the underlying topological space of any [[scheme (mathematics)|scheme]]. 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]]. Such spaces naturally occur in [[computer science]], specifically in [[denotational semantics]].

== 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 set]] that contains one of these points and not the other. More precisely the topological space ''X'' is Kolmogorov or <math>\mathbf T_0</math> if and only if:<ref name=":0" />

:If <math>a,b\in X</math> and <math>a\neq b</math>, there exists an open set ''O'' such that either <math>(a\in O) \wedge (b\notin O)</math> or <math>(a\notin O) \wedge (b\in O)</math>.

Note that topologically distinguishable points are automatically distinct. On the other hand, if the [[singleton set]]s {''x''} and {''y''} are [[separated sets|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]] they are distinguishable. This is how the T0 axiom fits in with the rest of the [[separation axiom]]s.

==Examples and counter examples==

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

===Spaces that are not T0===

*A set with more than one element, with the [[trivial topology]]. 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]] 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]]s ''f'' from the [[real line]] '''R''' to the [[complex plane]] '''C''' such that the [[Lebesgue integral]] <math>\left(\int_{\mathbb{R}} |f(x)|^2 \,dx\right)^{\frac{1}{2}} < \infty </math>. Two functions which are equal [[almost everywhere]] are indistinguishable. See also below.

===Spaces that are T0 but not T1===

*The [[Zariski topology]] on Spec(''R''), the [[prime spectrum]] of a [[commutative ring]] ''R'', is always T0 but generally not T1. The non-closed points correspond to [[prime ideal]]s which are not [[maximal ideal|maximal]]. They are important to the understanding of [[scheme (mathematics)|scheme]]s.
*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 space]] which is the particular point topology on the set {0,1}.
*The [[excluded point topology]] on any set with at least two elements is T0 but not T1. The only closed point is the excluded point.
*The [[Alexandrov topology]] on a [[partially 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.
*The [[right order topology]] on a [[totally ordered set]] 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 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 ==

Commonly studied topological spaces are all T0.
Indeed, when mathematicians in many fields, notably [[analysis (mathematics)|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 [[Lp 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 set|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 [[0 (number)|zero]].
The standard solution is to define L2('''R''') to be a set of [[equivalence class]]es of functions instead of a set of functions directly.
This constructs a [[Quotient space (topology)|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]]. No matter what topological space ''X'' might be to begin with, the [[Quotient space (topology)|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 (category theory)|natural]]ly [[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 [[structure (mathematics)|structure]]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]], and it has a seminorm, and these define a [[pseudometric space|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 space|complete]]. The space is not T0 since any two functions in L2('''R''') that 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 class]]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 space|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 space|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 space|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 space|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.

== See also ==

* [[Sober space]]

==References==
{{Reflist}}
*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).

[[Category:Separation axioms]]
[[Category:Properties of topological spaces]]

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Kolmogorov space