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.

If the intuition behind open subsets is that you can take a point in an open subset, move it a small distance, and get a point in the open subset; then an analogous intuition behind entourages is that you can take any two points related by an entourage, move them anywhere in the space as long as each is only moved a small distance relative to the other, and get two points related by the entourage.

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 $ϵ$ such that

  $$d(x,y)<ϵ\Rightarrow x\approx y,$$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 $ϵ$ 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

  $$x/y\in N\Rightarrow x{\approx }_{N}y,$$x/y \in N \;\implies\; x \approx_N 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.