Normal 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 the ball of radius $\alpha$ around~$x$. We introduce the class of normal ball structures as an asymptotic counterpart of normal topological spaces. The part of continuous functions in this situation is played by the slowly oscillation functions. We describe the ball analogues of pseudocompactness and discreteness and define a corona of a normal ball structure which is a generalization of the Higson corona of a~proper metric space.