Ultrafilters on metric Spaces

Let $X$ be an unbounded metric space, $B(x,r) = {y\in X: d(x,y) \leqslant r}$forall$x\in X$and$r\geqslant 0$\mathrm{.}Weendow$X$ with the discrete topology and identify the Stone-\v{C}ech 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-\v{C}ech compactification of a discrete semigroup \cite{b6}. We denote X^# = \{p\in \beta X: eachP\in pis unbounded inX\} and, for p,q \in X^#, write $p\parallel q$ if and only if there is $r \geqslant 0$suchthat$B(Q,r)\in p$foreach$Q\in q$,where$B(Q, r)=\cup_{x\in Q}B(x,r)$\mathrm{.}Asubset$S\subseteq X^#$iscalledinvariantif$p\in S$and$q\parallel p$imply$q\in S$. We characterize the minimal closed invariant subsets of $X$, the closure of the set $K(X^#) = \bigcup{M : M\mbox{is a minimal closed invariant subset of}X^#}$, and find the number of all minimal closed invariant subsets of X^#. For a subset $Y\subseteq X$ and p\in X^#, we denote $\bigtriangleup_p(Y) = Y^# \cap {q\in X^#: p \parallel q}$andsaythatasubset$S\subseteq X^#$ is an ultracompanion of $Y$ if $S={△}_p(Y)$ for some p\in X^#. We characterize large, thick, prethick, small, thin and asymptotically scattered spaces in terms of their ultracompanions.