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 , a neighbourhood of defines a generalized notion of distance from ; a point is [less than] this distance from iff . Uniformizing this, an entourage defines a generalized notion of uniform distance; two points and are [less than] this distance from each other iff . Or said another way, if is a neighbourhood of , then any point that is sufficiently close to will be in ; if is an entourage, then any two points and that are sufficiently close together will be related by .
The precise definition depends on the context.
where are points in the metric space and is the metric.
In a gauge space, is an entourage if there exists an and a gauging distance such that the preceding condition holds.
where 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 in place of .
In nonstandard analysis, every point in a topological space has an infinitesimal neighbourhood in the nonstandard extension , called the halo (or monad) of . This is to be thought of as the set of all of the hyperpoint?s that are infinitely close (adequal?) to . Similarly, any uniform space has an infinitesimal entourage in its nonstandard extension , a binary relation that relates two hyperpoints iff they are infinitely close to each other. (So the infinitesimal entourage is simply the adequality? relation.)