nLab Kolmogorov topological space




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


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A topological space (X,τ)(X,\tau) is called a Kolmogorov space if it satisfies the T 0T_0-separation axiom, hence if for x 1x 2Xx_1 \neq x_2 \in X any two distinct points, then at least one of them has an open neighbourhood U x iτU_{x_i} \in \tau which does not contain the other point. This is equivalent to its contrapositive: for all points x 1,x 2Xx_1, x_2 \in X, if every open neighbourhood U x iτU_{x_i} \in \tau which contains one of the points also contains the other point, then the two points are equal x 1=x 2x_1 = x_2.

the main separation axioms

T 0T_0Kolmogorovgiven two distinct points, at least one of them has an open neighbourhood not containing the other pointevery irreducible closed subset is the closure of at most one point
T 1T_1given two distinct points, both have an open neighbourhood not containing the other pointall points are closed
T 2T_2Hausdorffgiven two distinct points, they have disjoint open neighbourhoodsthe diagonal is a closed map
T >2T_{\gt 2}T 1T_1 and…all points are closed and…
T 3T_3regular Hausdorff…given a point and a closed subset not containing it, they have disjoint open neighbourhoods…every neighbourhood of a point contains the closure of an open neighbourhood
T 4T_4normal Hausdorff…given two disjoint closed subsets, they have disjoint open neighbourhoods…every neighbourhood of a closed set also contains the closure of an open neighbourhood
… every pair of disjoint closed subsets is separated by an Urysohn function

In dependent type theory

In dependent type theory, given a type XX, the type of subtypes of XX is the function type XΩX \to \Omega, where Ω\Omega is the type of all propositions with the type reflector type family P:ΩEl Ω(P)typeP:\Omega \vdash \mathrm{El}_\Omega(P) \; \mathrm{type}.

Given a type universe Type\mathrm{Type}, a topological space is a type XX with a type of subtypes O(X)O(X) with canonical embedding i O:O(X)(XΩ)i_O:O(X) \hookrightarrow (X \to \Omega), called the open sets of XX, which are closed under finite intersections and Type\mathrm{Type}-small unions.

Given a topological space (X,O(X))(X, O(X)), we define the relation

x:X,U:O(X)xUtypex:X, U:O(X) \vdash x \in U \; \mathrm{type}


xUEl Ω((i O(U))(x))x \in U \coloneqq \mathrm{El}_\Omega((i_O(U))(x))

By definition of the type of all propositions and its type reflector, xUx \in U is always a h-proposition for all x:Xx:X and U:O(X)U:O(X).

The specialization order is given by the type

U:O(X)(xU)(yU)\prod_{U:O(X)} (x \in U) \to (y \in U)

and for all elements x:Xx:X, there is an element

refl SpecOrd(x): U:O(X)(xU)(xU)\mathrm{refl}_{SpecOrd}(x):\prod_{U:O(X)} (x \in U) \to (x \in U)

defined by the identity function on xUx \in U

refl SpecOrd(x,U)id xU\mathrm{refl}_{SpecOrd}(x, U) \coloneqq \mathrm{id}_{x \in U}

Since xUx \in U and yUy \in U are both h-propositions, and h-propositions are closed under function types and dependent product types, the specialization order is also valued in h-propositions.

There is a canonical family of functions for every topological space

x:X,y:XidToSpecOrd(x,y):(x= Xy)( U:O(X)(xU)(yU))×( U:O(X)(yU)(xU))x:X, y:X \vdash \mathrm{idToSpecOrd}(x, y):(x =_X y) \to \left(\prod_{U:O(X)} (x \in U) \to (y \in U)\right) \times \left(\prod_{U:O(X)} (y \in U) \to (x \in U)\right)

inductively defined on reflexivity of the identity type by

idToSpecOrd(x,x)(id X(x))refl SpecOrd(x)\mathrm{idToSpecOrd}(x, x)(\mathrm{id}_X(x)) \coloneqq \mathrm{refl}_{SpecOrd}(x)


A Kolmogorov topological space or T 0T_0-space is a topological space which satisfies the T 0T_0-separation axiom: idToSpecOrd(x,y)\mathrm{idToSpecOrd}(x, y) is an equivalence of types for all x:Xx:X and y:Xy:X.

T 0: x:X y:XisEquiv(idToSpecOrd(x,y))T_0:\prod_{x:X} \prod_{y:X} \mathrm{isEquiv}(\mathrm{idToSpecOrd}(x, y))

Since the specialization order is valued in propositions, the T 0T_0-separation axiom ensures that the topological space is an h-set.


Alternative Characterizations

Regard Sierpinski space Σ\Sigma as a frame object in the category of topological spaces (meaning: the representable functor Top(,Σ):Top opSetTop(-, \Sigma): Top^{op} \to Set lifts through the monadic forgetful functor FrameSetFrame \to Set), hence as a dualizing object that induces a contravariant adjunction between frames and topological spaces. Thus for a space XX, the frame Top(X,Σ)Top(X, \Sigma) is the frame of open sets; for a frame AA, there is an accompanying topological space of points Frame(A,Σ)Frame(A, \Sigma), a subspace of the product space Σ |A|\Sigma^{{|A|}}. The unit of the adjunction is called the double dual embedding.


A topological space XX is T 0T_0 precisely when the double dual embedding XFrame(Top(X,Σ),Σ)X \to Frame(Top(X, \Sigma), \Sigma) is a monomorphism.

The proof is trivial: the monomorphism condition translates to saying that for any points x,yXx, y \in X, if the truth values of xUx \in U and yUy \in U agree for every open set UU, then x=yx = y.

In terms of lifting properties

The separation conditions T 0T_0 to T 4T_4 may equivalently be understood as lifting properties against certain maps of finite topological spaces, among others.

This is discussed at separation axioms in terms of lifting properties, to which we refer for further details. Here we just briefly indicate the corresponding lifting diagrams.

In the following diagrams, the relevant finite topological spaces are indicated explicitly by illustration of their underlying point set and their open subsets:

  • points (elements) are denoted by \bullet with subscripts indicating where the points map to;

  • boxes are put around open subsets,

  • an arrow u c\bullet_u \to \bullet_c means that c\bullet_c is in the topological closure of u\bullet_u.

In the lifting diagrams for T 2T 4T_2-T_4 below, an arrow out of the given topological space XX is a map that determines (classifies) a decomposition of XX into a union of subsets with properties indicated by the picture of the finite space.

Notice that the diagrams for T 2T_2-T 4T_4 below do not in themselves imply T 1 T_1 .


(Lifting property encoding T 0T_0)
The following lifting property in Top equivalently encodes the separation axiom T 0 T_0 :


(Lifting property encoding T 1T_1)
The following lifting property in Top equivalently encodes the separation axiom T 1 T_1 :


(Lifting property encoding T 2T_2)
The following lifting property in Top equivalently encodes the separation axiom T 2 T_2 :


(Lifting property encoding T 3T_3)
The following lifting property in Top equivalently encodes the separation axiom T 3 T_3 :


(Lifting property encoding T 4T_4)
The following lifting property in Top equivalently encodes the separation axiom T 4 T_4 :



(Kolmogorov quotient)

Let (X,τ)(X,\tau) be a topological space. Consider the relation on the underlying set by which x 1x 1x_1 \sim x_1 precisely if neither x ix_i has an open neighbourhood not containing the other. This is an equivalence relation. The quotient topological space XX/X \to X/\sim by this equivalence relation is a T 0T_0-space.

This construction is the reflector exhibiting Kolmogorov spaces as a reflective subcategory of the category Top of all topological spaces.

preorderpartial orderequivalence relationequality
topological spaceKolmogorov topological spacesymmetric topological spaceaccessible topological space?


Last revised on May 25, 2023 at 14:32:05. See the history of this page for a list of all contributions to it.