nLab
entourage

Entourages

Idea

An entourage (aka vicinity) is a binary relation of ‘approximate equality’ on a space, generally a uniform space. Just as a topological space is given by its underlying set of points and an appropriate collection of open subsets, so a uniform space is given by its underlying set of points and an appropriate collection of entourages.

Entourages are actually a uniformization of neighbourhoods rather than of open sets as such. Given a point xx, a neighbourhood UU of xx defines a generalized notion of distance from xx; a point yy is [less than] this distance from xx iff yUy \in U. Uniformizing this, an entourage EE defines a generalized notion of uniform distance; two points xx and yy are [less than] this distance from each other iff (x,y)E(x,y) \in E. Or said another way, if UU is a neighbourhood of xx, then any point yy that is sufficiently close to xx will be in UU; if EE is an entourage, then any two points xx and yy that are sufficiently close together will be related by EE.

Definitions

The precise definition depends on the context.

  • In a metric space, a relation \approx is an entourage if there exists a positive real number ϵ\epsilon such that

    d(x,y)<ϵxy, d(x,y) \lt \epsilon \;\implies\; x \approx y ,

    where x,yx,y are points in the metric space and dd is the metric.

  • In a gauge space, \approx is an entourage if there exists an ϵ\epsilon and a gauging distance dd such that the preceding condition holds.

  • In a topological abelian group, \approx is an entourage if there is a neighbourhood NN of the identity element such that

    x/yNxy, x/y \in N \;\implies\; x \approx y ,

    where x,yx,y are points in the metric space and // is the division operation in the group.

  • In a nonabelian topological group, there are two distinct notions of entourage, one using the same formula as above and the other using y/xy/x in place of x/yx/y.

  • Of course, the most general kind of entourage is that occurring in the definition of a uniform space, in the same way that open sets occur in the definition of a topological space.

Infinitesimal entourages

In nonstandard analysis, every point xx in a topological space XX has an infinitesimal neighbourhood in the nonstandard extension X *X^*, called the halo (or monad) of xx. This is to be thought of as the set {yX *|yx}\{ y \in X^* \;|\; y \simeq x \} of all of the hyperpoint?s that are infinitely close (adequal?) to xx. Similarly, any uniform space XX has an infinitesimal entourage in its nonstandard extension X *X^*, a binary relation {xX *,yX *|yx}\{ x \in X^*,\; y \in X^* \;|\; y \simeq x \} that relates two hyperpoints iff they are infinitely close to each other. (So the infinitesimal entourage is simply the adequality? relation.)

Revised on December 16, 2016 01:17:54 by Toby Bartels (108.167.41.14)