Uniform ball structures

A ball structure is a triple $\mathbb B=(X,P,B)$, where $X,P$ are nonempty sets and, for all $x\in X$, $\alpha \in P$, $B(x,\alpha )$ is a subset of $X, x\in B(x,\alpha)$, which is called a ball of radius $\alpha$ around $x$. We introduce the class of uniform ball structures as an asymptotic counterpart of the class of uniform topological spaces. We show that every uniform ball structure can be approximated by metrizable ball structures. We also define two types of ball structures closed to being metrizable, and describe the extremal elements in the classes of ball structures with fixed support $X$.