{"paper":{"title":"Fibers and local connectedness of planar continua","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Beno\\^it Loridant, Jun Luo","submitted_at":"2017-03-17T07:39:55Z","abstract_excerpt":"We describe non-locally connected planar continua via the concepts of fiber and numerical scale.\n  Given a continuum $X\\subset\\mathbb{C}$ and $x\\in\\partial X$, we show that the set of points $y\\in \\partial X$ that cannot be separated from $x$ by any finite set $C\\subset \\partial X$ is a continuum. This continuum is called the {\\em modified fiber} $F_x^*$ of $X$ at $x$. If $x\\in X^o$, we set $F^*_x=\\{x\\}$. For $x\\in X$, we show that $F_x^*=\\{x\\}$ implies that $X$ is locally connected at $x$. We also give a concrete planar continuum $X$, which is locally connected at a point $x\\in X$ while the f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.05914","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"}