Resolvability of ball structures

A ball structure is a triple B = (X, P, B) where X, P are nonempty sets and, for any x ∈ X, α ∈ P, B(x, α) is a subset of X, which is called a ball of radius α around x. It is supposed that x ∈ B(x, α) for any x ∈ X, α ∈ P. A subset Y C X is called large if X = B(Y, α) for some α ∈ P where B(Y, α) = Uy∈Y B(y, α). The set X is called a support of B, P is called a set of radiuses. Given a cardinal k, B is called k-resolvable if X can be partitioned to k large subsets. The cardinal res B = sup {k : B is k-resolvable} is called a resolvability of B. We determine the resolvability of the ball structures related to metric spaces, groups and filters