{"paper":{"title":"Ultrafilters on metric Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"I.V. Protasov","submitted_at":"2013-10-09T11:36:08Z","abstract_excerpt":"Let $X$ be an unbounded metric space, $B(x,r) = \\{y\\in X: d(x,y) \\leqslant r\\}$ for all $x\\in X$ and $r\\geqslant 0$. We endow $X$ 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}.\n  We denote $X^# = \\{p\\in \\beta X: \\mbox{each}P\\in p\\mbox{is unbounded in}X\\}$ and, for $p,q \\in X^#$, write $p\\parallel q$ if and only if there is $r \\geqslant 0$ such that $B(Q,r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2437","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}