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We show that under the interpretation of the modality $\\Diamond$ as the derived set (of limit points) operation, K4$\\mathbb{C}_n$ is characterised by validity in all spaces that are hereditarily $n+1$-irresolvable and have the T$_D$ separation property.\n  We also"},"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":"2003.12946","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2020-03-29T04:34:38Z","cross_cats_sorted":[],"title_canon_sha256":"13e772766441c5300e43de5f2ea94fc8c50772d1df4425c22f73ba5ddb6f948c","abstract_canon_sha256":"59bf8c89f1081eaeaf90712511100b4e150d5ab2f5be5317d3075d62f5ea5370"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:09:41.057777Z","signature_b64":"99jlA4aoLCsRxydQ20/JDaRt874O0Td3+iQik82aPqTWV+cenXV44ywS+3ZZi2kYJNEq2TXtAnjQZYagngMSCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45751dbe18186587210e23c1a77bbd673b190c01e6a5702bf63c98a111e700b5","last_reissued_at":"2026-07-05T07:09:41.057302Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:09:41.057302Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Modal Logics of Some Hereditarily Irresolvable Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Robert Goldblatt","submitted_at":"2020-03-29T04:34:38Z","abstract_excerpt":"A topological space is \\emph{hereditarily $k$-irresolvable} if none of its subspaces can be partitioned into $k$ dense subsets, We use this notion to provide a topological semantics for a sequence of modal logics whose $n$-th member K4$\\mathbb{C}_n$ is characterised by validity in transitive Kripke frames of circumference at most $n$. 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