{"paper":{"title":"Directions in Type I spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Mathieu Baillif","submitted_at":"2014-04-04T21:32:47Z","abstract_excerpt":"A direction in a Type I space $X=\\cup_{\\alpha<\\omega_1}X_\\alpha$ is a closed and unbounded subset $D$ of $X$ such that given any continuous $f:X\\to\\mathbb{L}_{\\ge 0}$ (the closed long ray), if $f$ is unbounded on $D$ then $f$ is unbounded on each unbounded subset of $D$. A closed copy of $\\omega_1$ is a direction in any Type I space. We study various aspects of directions and show some independence results. A sample: There is an $\\omega$-bounded Type I space without direction; PFA implies that a locally compact countably tight $\\omega_1$-compact Type I space contains a direction; if there is a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.1398","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"}