{"paper":{"title":"A remark on locally direct product subsets in a topological Cartesian space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Hiroki Yagisita","submitted_at":"2019-08-15T16:26:07Z","abstract_excerpt":"Let $X$ and $Y$ be topological spaces. Let $C$ be a path-connected closed set of $X\\times Y$. Suppose that $C$ is locally direct product, that is, for any $(a,b)\\in X\\times Y$, there exist an open set $U$ of $X$, an open set $V$ of $Y$, a subset $I$ of $U$ and a subset $J$ of $V$ such that $(a,b) \\in U\\times V$ and $$C\\cap (U\\times V)=I\\times J$$ hold. Then, in this memo, we show that $C$ is globally so, that is, there exist a subset $A$ of $X$ and a subset $B$ of $Y$ such that $$C=A\\times B$$ holds. The proof is elementary. Here, we note that one might be able to think of a (perhaps, open) si"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05624","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.05624/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}