Ultrafilters on metric spaces

Let X be an unbounded metric space, B(x,r) = fy is an element of X: d(x,y) r} for all x is an element of X and r 0. We endow X with the discrete topology and identify the StoneOech compactification beta X of X with the set of all ultrafilters on X. Our aim is to reveal some features of algebra in beta X similar to the algebra in the Stone-Oech compactification of a discrete semigroup [6]. We denote X-# = {p is an element of beta X: each P is an element of p is unbounded in X} and, for p, q is an element of X-#, write p parallel to q if and only if there is r >= 0 such that B(Q,r) is an element of p for each Q is an element of q, where B(Q,r) = U. EQ B(x,r). A subset S C X# is called invariant if p is an element of S and q p imply q is an element of S. We characterize the minimal closed invariant subsets of X, the closure of the set K(X-#) = LAM: M is a minimal closed invariant subset of X-#}, and find the number of all minimal closed invariant subsets of X. For a subset Y subset of X and p is an element of X-#, we denote Delta p(Y) = Y-#boolean AND{q is an element of XO: p parallel to q} and say that a subset S subset of X-# is an ultracompanion of Y if S = Delta(p) (Y) for some p is an element of X-#. We characterize large, thick, prethick, small, thin and asymptotically scattered spaces in terms of their ultracompanions. (C) 2014 Elsevier B.V. All rights reserved.