# Topological indistinguishability

In topology, two points of a topological space *X* are **topologically indistinguishable** if they have exactly the same neighborhoods. That is, if *x* and *y* are points in *X*, and *N _{x}* is the set of all neighborhoods that contain

*x*, and

*N*is the set of all neighborhoods that contain

_{y}*y*, then

*x*and

*y*are "topologically indistinguishable" if and only if

*N*=

_{x}*N*. (See Hausdorff's axiomatic neighborhood systems.)

_{y}Intuitively, two points are topologically indistinguishable if the topology of *X* is unable to discern between the points.

Two points of *X* are **topologically distinguishable** if they are not topologically indistinguishable. This means there is an open set containing precisely one of the two points (equivalently, there is a closed set containing precisely one of the two points). This open set can then be used to distinguish between the two points. A **T _{0} space** is a topological space in which every pair of distinct points is topologically distinguishable. This is the weakest of the separation axioms.

Topological indistinguishability defines an equivalence relation on any topological space *X*. If *x* and *y* are points of *X* we write *x* ≡ *y* for "*x* and *y* are topologically indistinguishable". The equivalence class of *x* will be denoted by [*x*].

## Examples

For T_{0} spaces (in particular, for Hausdorff spaces) the notion of topological indistinguishability is trivial, so one must look to non-T_{0} spaces to find interesting examples. On the other hand, regularity and normality do not imply T_{0}, so we can find examples with these properties. In fact, almost all of the examples given below are completely regular.

- In an indiscrete space, any two points are topologically indistinguishable.
- In a pseudometric space, two points are topologically indistinguishable if and only if the distance between them is zero.
- In a seminormed vector space,
*x*≡*y*if and only if ‖*x*−*y*‖ = 0.- For example, let
*L*^{2}(**R**) be the space of all measurable functions from**R**to**R**which are square integrable (see*L*^{p}space). Then two functions*f*and*g*in*L*^{2}(**R**) are topologically indistinguishable if and only if they are equal almost everywhere.

- For example, let
- In a topological group,
*x*≡*y*if and only if*x*^{−1}*y*∈ cl{*e*} where cl{*e*} is the closure of the trivial subgroup. The equivalence classes are just the cosets of cl{*e*} (which is always a normal subgroup). - Uniform spaces generalize both pseudometric spaces and topological groups. In a uniform space,
*x*≡*y*if and only if the pair (*x*,*y*) belongs to every entourage. The intersection of all the entourages is an equivalence relation on*X*which is just that of topological indistinguishability. - Let
*X*have the initial topology with respect to a family of functions . Then two points*x*and*y*in*X*will be topologically indistinguishable if the family does not separate them (i.e. for all ). - Given any equivalence relation on a set
*X*there is a topology on*X*for which the notion of topological indistinguishability agrees with the given equivalence relation. One can simply take the equivalence classes as a base for the topology. This is called the partition topology on*X*.

## Specialization preorder

The topological indistinguishability relation on a space *X* can be recovered from a natural preorder on *X* called the specialization preorder. For points *x* and *y* in *X* this preorder is defined by

*x*≤*y*if and only if*x*∈ cl{*y*}

where cl{*y*} denotes the closure of {*y*}. Equivalently, *x* ≤ *y* if the neighborhood system of *x*, denoted *N*_{x}, is contained in the neighborhood system of *y*:

*x*≤*y*if and only if*N*_{x}⊂*N*_{y}.

It is easy to see that this relation on *X* is reflexive and transitive and so defines a preorder. In general, however, this preorder will not be antisymmetric. Indeed, the equivalence relation determined by ≤ is precisely that of topological indistinguishability:

*x*≡*y*if and only if*x*≤*y*and*y*≤*x*.

A topological space is said to be symmetric (or R_{0}) if the specialization preorder is symmetric (i.e. *x* ≤ *y* implies *y* ≤ *x*). In this case, the relations ≤ and ≡ are identical. Topological indistinguishability is better behaved in these spaces and easier to understand. Note that this class of spaces includes all regular and completely regular spaces.

## Properties

### Equivalent conditions

There are several equivalent ways of determining when two points are topologically indistinguishable. Let *X* be a topological space and let *x* and *y* be points of *X*. Denote the respective closures of *x* and *y* by cl{*x*} and cl{*y*}, and the respective neighborhood systems by *N*_{x} and *N*_{y}. Then the following statements are equivalent:

*x*≡*y*- for each open set
*U*in*X*,*U*contains either both*x*and*y*or neither of them *N*_{x}=*N*_{y}*x*∈ cl{*y*} and*y*∈ cl{*x*}- cl{
*x*} = cl{*y*} *x*∈ ∩*N*_{y}and*y*∈ ∩*N*_{x}- ∩
*N*_{x}= ∩*N*_{y} *x*∈ cl{*y*} and*x*∈ ∩*N*_{y}*x*belongs to every open set and every closed set containing*y*- a net or filter converges to
*x*if and only if it converges to*y*

These conditions can be simplified in the case where *X* is symmetric space. For these spaces (in particular, for regular spaces), the following statements are equivalent:

*x*≡*y*- for each open set
*U*, if*x*∈*U*then*y*∈*U* *N*_{x}⊂*N*_{y}*x*∈ cl{*y*}*x*∈ ∩*N*_{y}*x*belongs to every closed set containing*y**x*belongs to every open set containing*y*- every net or filter that converges to
*x*converges to*y*

### Equivalence classes

To discuss the equivalence class of *x*, it is convenient to first define the upper and lower sets of *x*. These are both defined with respect to the specialization preorder discussed above.

The lower set of *x* is just the closure of {*x*}:

while the upper set of *x* is the intersection of the neighborhood system at *x*:

The equivalence class of *x* is then given by the intersection

Since ↓*x* is the intersection of all the closed sets containing *x* and ↑*x* is the intersection of all the open sets containing *x*, the equivalence class [*x*] is the intersection of all the open and closed sets containing *x*.

Both cl{*x*} and ∩*N*_{x} will contain the equivalence class [*x*]. In general, both sets will contain additional points as well. In symmetric spaces (in particular, in regular spaces) however, the three sets coincide:

In general, the equivalence classes [*x*] will be closed if and only if the space is symmetric.

### Continuous functions

Let *f* : *X* → *Y* be a continuous function. Then for any *x* and *y* in *X*

*x*≡*y*implies*f*(*x*) ≡*f*(*y*).

The converse is generally false (There are quotients of T_{0} spaces which are trivial). The converse will hold if *X* has the initial topology induced by *f*. More generally, if *X* has the initial topology induced by a family of maps then

*x*≡*y*if and only if*f*_{α}(*x*) ≡*f*_{α}(*y*) for all α.

It follows that two elements in a product space are topologically indistinguishable if and only if each of their components are topologically indistinguishable.

## Kolmogorov quotient

Since topological indistinguishability is an equivalence relation on any topological space *X*, we can form the quotient space *KX* = *X*/≡. The space *KX* is called the **Kolmogorov quotient** or **T _{0} identification** of

*X*. The space

*KX*is, in fact, T

_{0}(i.e. all points are topologically distinguishable). Moreover, by the characteristic property of the quotient map any continuous map

*f*:

*X*→

*Y*from

*X*to a T

_{0}space factors through the quotient map

*q*:

*X*→

*KX*.

Although the quotient map *q* is generally not a homeomorphism (since it is not generally injective), it does induce a bijection between the topology on *X* and the topology on *KX*. Intuitively, the Kolmogorov quotient does not alter the topology of a space. It just reduces the point set until points become topologically distinguishable.