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Equivalently for distinct x, y in X there are open sets U,V with (x,y) in U x V and either U x V a subset of E or U x V a subset of E complement.\n  For clopen graphs E_1 and E_2 on the Baire space (omega^omega) we say that E_1 continuously reduces to E_2 iff there is a continuous map f from the Baire space to itself such that for\n  [(x,y) in E_1 iff (f(x),f(y)) in E_2 ] for distinct x,y. Note that f need not be one-to-one but there should be no e"},"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":"1210.8362","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-10-31T15:20:35Z","cross_cats_sorted":[],"title_canon_sha256":"49d344ae7761427252a1c7cc96ef2f71a9bddc9c66f10c0f29cb28a3eb374b37","abstract_canon_sha256":"bae601b2b3ea7ffc41c25afacafa101f32092e9d3ff9a014b6f209775473b196"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:55.255397Z","signature_b64":"NR7hp43M7JbC1BA7i++vIDBjYaMYrp25p5YM94dJZp9YCpFR7i2xHq2US1FehFTp1m1tjbnKbTV8thQBNDnWBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ce2394cca0030b6b0e316ef1f06436eb1ca4a382ce4fa76565f253326f06724","last_reissued_at":"2026-05-18T03:41:55.254589Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:55.254589Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A hierarchy of clopen graphs on the Baire space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Arnold W. Miller","submitted_at":"2012-10-31T15:20:35Z","abstract_excerpt":"We say that binary relation E on a space X is a clopen graph on X iff E is symmetric and irreflexive and clopen relative to X x X minus its diagonal. Equivalently for distinct x, y in X there are open sets U,V with (x,y) in U x V and either U x V a subset of E or U x V a subset of E complement.\n  For clopen graphs E_1 and E_2 on the Baire space (omega^omega) we say that E_1 continuously reduces to E_2 iff there is a continuous map f from the Baire space to itself such that for\n  [(x,y) in E_1 iff (f(x),f(y)) in E_2 ] for distinct x,y. 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