nLab Kolmogorov topological space

Definition

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

the main separation axioms

number	name	statement	reformulation
$T_0$	Kolmogorov	given two distinct points, at least one of them has an open neighbourhood not containing the other point	every irreducible closed subset is the closure of at most one point
$T_1$		given two distinct points, both have an open neighbourhood not containing the other point	all points are closed
$T_2$	Hausdorff	given two distinct points, they have disjoint open neighbourhoods	the diagonal is a closed map
$T_{\gt 2}$		$T_1$ and…	all points are closed and…
$T_3$	regular 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
$T_4$	normal 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 $X$, the type of subtypes of $X$ is the function type $X \to \Omega$, where $\Omega$ is the type of all propositions with the type reflector type family $P:\Omega \vdash \mathrm{El}_\Omega(P) \; \mathrm{type}$.

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

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

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

by

$$x \in U \coloneqq \mathrm{El}_\Omega((i_O(U))(x))$$

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

The specialization order is given by the type

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

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

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

defined by the identity function on $x \in U$

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

Since $x \in U$ and $y \in U$ 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

$$x:X, y:X \vdash \mathrm{idToSpecOrd}(x, y):(x =_X y) \to \left(\prod_{U:O(X)} (x \in U) \to (y \in U)\right) \times \left(\prod_{U:O(X)} (y \in U) \to (x \in U)\right)$$

inductively defined on reflexivity of the identity type by

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

Definition

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

$$T_0:\prod_{x:X} \prod_{y:X} \mathrm{isEquiv}(\mathrm{idToSpecOrd}(x, y))$$

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

Properties

Alternative Characterizations

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

Proposition

A topological space $X$ is $T_0$ precisely when the double dual embedding $X \to Frame(Top(X, \Sigma), \Sigma)$ is a monomorphism.

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

In terms of lifting properties

The separation conditions $T_0$ to $T_4$ 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 $\bullet_u \to \bullet_c$ means that $\bullet_c$ is in the topological closure of $\bullet_u$.

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

Notice that the diagrams for $T_2$-$T_4$ below do not in themselves imply $T_1$.

Proposition

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

Proposition

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

Proposition

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

Proposition

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

Proposition

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

Reflection

Proposition

(Kolmogorov quotient)

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

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

Related concepts

• Hausdorff topological space

• nice topological space

preorder	partial order	equivalence relation	equality
topological space	Kolmogorov topological space	symmetric topological space	accessible topological space?

References

• Wikipedia, Kolmogorov space