{"paper":{"title":"Networks for the weak topology of Banach and Fr\\'echet spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"J. K\\c{a}kol, S. Gabriyelyan, W. Kubi\\'s, W. Marciszewski","submitted_at":"2014-12-04T18:02:53Z","abstract_excerpt":"We start the systematic study of Fr\\'{e}chet spaces which are $\\aleph$-spaces in the weak topology. A topological space $X$ is an $\\aleph_0$-space or an $\\aleph$-space if $X$ has a countable $k$-network or a $\\sigma$-locally finite $k$-network, respectively. We are motivated by the following result of Corson (1966): If the space $C_{c}(X)$ of continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology is a Banach space, then $C_{c}(X)$ endowed with the weak topology is an $\\aleph_0$-space if and only if $X$ is countable. We extend Corson's result as follows"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.1748","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"}