Coronas of Balleans
by
I. Protasov
Kiev, Ukraine

A ballean is a triple (X, R, B) where X, R are nonempty sets and, for all x from X, r from R, B(x, r) is a subset of X which is called a ball of radius r around x.

Let (X₁, R₁, B₁), (X₂, R₂, B₂) be balleans. A mapping f from X₁ to X₂ is called coarse if, for every r₁ from R₁ there exists r₂ from R₂ such that f(B₁(x, r₁)) is contained in B₂(f(x), r₂) for every x from X. A bijection f is called an isomorphism if f and its inverse are coarse. We show that the balleans (with the isomorphisms defined above) can be considered as the asymptotic counterparts of uniform topological spaces. On the ballean stage the part of continuous functions play slow oscilating functions. We define a corona of ballean as a generalization of Higson's corona and a counterpart of Stone-Cech compactification.

Date received: May 26, 2003