{"paper":{"title":"Unifying large scale and small scale geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN","math.GT"],"primary_cat":"math.MG","authors_text":"Jerzy Dydak","submitted_at":"2018-03-24T19:24:17Z","abstract_excerpt":"A topology on a set $X$ is the same as a projection (i.e. an idempotent linear operator) $cl:2^X\\to 2^X$ satisfying $A\\subset cl(A)$ for all $A\\subset X$. That's a good way to summarize Kuratowski's closure operator.\n  Basic geometry on a set $X$ is a dot product $\\cdot:2^X\\times 2^X\\to 2^Y$. Its equivalent form is an orthogonality relation on subsets of $X$. The optimal case is if the orthogonality relation satisfies a variant of parallel-perpendicular decomposition from linear algebra.\n  We show that this concept unifies small scale (topology, proximity spaces, uniform spaces) and large scal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.09154","kind":"arxiv","version":2},"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"}