nLab uniform cover

Idea

A uniform cover is an open cover in which enough of the subsets have a uniform minimal size. This doesn't make sense in (say) an arbitrary topological space, but it does make sense in a uniform space (or a uniform locale). In fact, it's possible to define a uniform space by specifying which covers are uniform.

Definitions

The most general definition (that we have here so far) is in a uniform space or locale, but we can write down the definition in other familiar situations as well.

In a metric space

Let $X$ be a (pseudo)metric space. A collection $\mathcal{C}$ of subsets of $X$ is a uniform cover if:

• For some positive number $\epsilon$, every open ball of radius $\epsilon$ is contained in some element of $\mathcal{C}$.

In a topological group or vector space

Let $X$ be a topological vector space or even a topological abelian group. A collection $\mathcal{C}$ of subsets of $X$ is a uniform cover if:

• For some neighbourhood $N$ of $0$ (the identity element of the group), every set of the form $a + N$ is contained in some element of $\mathcal{C}$.

If $X$ is a topological group, then we have both left-uniform and right-uniform covers, depending on whether we use $a N$ or $N a$ in the definition above. (I'm not sure what the convention is, if any, on which is left and which is right.)

In a uniform space

Sometimes the notion of uniform cover is taken as axiomatic in the definition of uniform space. But if we define a uniform space in terms of entourages, then we have:

Let $X$ be a uniform space. A collection $\mathcal{C}$ of subsets of $X$ is a uniform cover if:

• For some entourage $U$, every set of the form $U[a] \coloneqq \{ b\colon X \;|\; a \approx_U b \}$ is contained in some element of $\mathcal{C}$.

This definition subsumes both (pseudo)metric spaces and topological groups (although a nonabelian topological group has two uniform structures, corresponding to the two notions of uniform cover).