Ultrafilters on metric spaces

Let X be an unbounded metric space, B ( x , r ) = { y ∈ X : d ( x , y ) ≤ r } for all x ∈ X and r ≥ 0 . We endow X with the discrete topology and identify the Stone–Čech compactification βX of X with the set of all ultrafilters on X. Our aim is to reveal some features of algebra in βX similar to the algebra in the Stone–Čech compactification of a discrete semigroup [6]. ote X # = { p ∈ β X : each P ∈ p is unbounded in X } and, for p , q ∈ X # , write p ∥ q if and only if there is r ≥ 0 such that B ( Q , r ) ∈ p for each Q ∈ q , where B ( Q , r ) = ⋃ x ∈ Q B ( x , r ) . A subset S ⊆ X # is called invariant if p ∈ S and q ∥ p imply q ∈ S . We characterize the minimal closed invariant subsets of X, the closure of the set K ( X # ) = ⋃ { M : M is a minimal closed invariant subset of X # } , and find the number of all minimal closed invariant subsets of X # . subset Y ⊆ X and p ∈ X # , we denote △ p ( Y ) = Y # ∩ { q ∈ X # : p ∥ q } and say that a subset S ⊆ X # is an ultracompanion of Y if S = △ p ( Y ) for some p ∈ X # . We characterize large, thick, prethick, small, thin and asymptotically scattered spaces in terms of their ultracompanions.