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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1412.1748","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-12-04T18:02:53Z","cross_cats_sorted":[],"title_canon_sha256":"f2d56386205e8249812e81fac80e7a3524c303c9f5a5e12b77d9f420720e4895","abstract_canon_sha256":"d89370912c466a3af5a6399344b0ec317fd18252e43612aeea4b1db3ce0da168"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:29:54.678886Z","signature_b64":"ddGsKKmId6mZjjBtCQng3gHVPZ+h6nBQhn7oqgeLsUaTPQP585Lfm/PIjCXt8PWWXsm4raUiJjyChon8BS0NCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63ee79e2586695baff4b7100e4ab8a379d024b3eba69292ed55048b0080ecca0","last_reissued_at":"2026-05-18T01:29:54.678246Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:29:54.678246Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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