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Theorem 4.4 then gives a new improvement of the well-known Hausdorff bound $2^{L(X)\\chi(X)}$ from which it follows that $|X|\\leq 2^{\\psi_c(X)}$ if $X$ is H-closed (Dow/Porter [5]). The invariant $aL^\\prime(X)$ is constructed using convergent open ultrafilters and an operator $c:\\scr{P}(X)\\to\\scr{P}(X)$ with the prope"},"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":"1610.09245","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2016-10-28T14:46:34Z","cross_cats_sorted":[],"title_canon_sha256":"4c68d48e7c816f1d72d89ffb83cca7d4fadeb092b15d83165d8ca604bf474634","abstract_canon_sha256":"674940e61e47863bbe2b817e949a2dbe988c45b089e9587be773561fb8ea87c0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:58.661343Z","signature_b64":"japOPk2tDz4gu4K7l8QNCV0CQxqct9GtOX9hGnClWNI2QtU02PDI3vezMAZl0g/HKxPu8gh4BHGPL/3drrr4Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"316d3c4d1db6fb7e0b7014285f1c4452101ac9db20a4104edc713c28e736af5d","last_reissued_at":"2026-05-18T01:00:58.660686Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:58.660686Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the cardinality of Hausdorff spaces and H-closed spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Jack Porter, Nathan Carlson","submitted_at":"2016-10-28T14:46:34Z","abstract_excerpt":"We introduce the cardinal invariant $aL^\\prime(X)$ and show that $|X|\\leq 2^{aL^\\prime(X)\\chi(X)}$ for any Hausdorff space $X$ (a corollary of Theorem 4.4. This invariant has the properties a) $aL^\\prime(X)=\\aleph_0$ if $X$ is H-closed, and b) $aL(X)\\leq aL^\\prime(X)\\leq aL_c(X)$. Theorem 4.4 then gives a new improvement of the well-known Hausdorff bound $2^{L(X)\\chi(X)}$ from which it follows that $|X|\\leq 2^{\\psi_c(X)}$ if $X$ is H-closed (Dow/Porter [5]). 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