{"paper":{"title":"Embeddings of finite-dimensional compacta in Euclidean spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"S. Bogataya, S. Bogatyi, V. Valov","submitted_at":"2010-10-10T03:59:17Z","abstract_excerpt":"If $g$ is a map from a space $X$ into $\\mathbb R^m$ and $q$ is an integer, let $B_{q,d,m}(g)$ be the set of all lines $\\Pi^d\\subset\\mathbb R^m$ such that $|g^{-1}(\\Pi^d)|\\geq q$. Let also $\\mathcal H(q,d,m,k)$ denote the maps $g\\colon X\\to\\mathbb R^m$ such that $\\dim B_{q,d,m}(g)\\leq k$. We prove that for any $n$-dimensional metric compactum $X$ each of the sets $\\mathcal H(3,1,m,3n+1-m)$ and $\\mathcal H(2,1,m,2n)$ is dense and $G_\\delta$ in the function space $C(X,\\mathbb R^m)$ provided $m\\geq 2n+1$ (in this case $\\mathcal H(3,1,m,3n+1-m)$ and $\\mathcal H(2,1,m,2n)$ can consist of embeddings)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.1892","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"}