nLab entourage

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 $x$, a neighbourhood $U$ of $x$ defines a generalized notion of distance from $x$; a point $y$ is [less than] this distance from $x$ iff $y \in U$. Uniformizing this, an entourage $E$ defines a generalized notion of uniform distance; two points $x$ and $y$ are [less than] this distance from each other iff $(x,y) \in E$. Or said another way, if $U$ is a neighbourhood of $x$, then any point $y$ that is sufficiently close to $x$ will be in $U$; if $E$ is an entourage, then any two points $x$ and $y$ that are sufficiently close together will be related by $E$.

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 $\epsilon$ such that

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

  where $x,y$ are points in the metric space and $d$ is the metric.

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

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

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

  where $x,y$ 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 $y/x$ in place of $x/y$.

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