{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T01:58:12Z","timestamp":1777687092408,"version":"3.51.4"},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2009,9,8]],"date-time":"2009-09-08T00:00:00Z","timestamp":1252368000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2009,10]]},"abstract":"<jats:p>The compact-open topology on the set of continuous functionals from the Baire space to the natural numbers is well known to be zero-dimensional. We prove that the closely related sequential topology on this set is not even regular. The sequential topology arises naturally as the topology carried by the exponential <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0960129509990065_char2\"\/><\/jats:private-char> formed in various cartesian closed categories of topological spaces. Moreover, we give an example of an effectively open subset of <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0960129509990065_char2\"\/><\/jats:private-char> that violates regularity. The topological properties of <jats:private-char><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0960129509990065_char2\"\/><\/jats:private-char> are known to be closely related to an open problem in Computable Analysis. We also show that the sequential topology on the space of continuous real-valued functions on a Polish space need not be regular.<\/jats:p>","DOI":"10.1017\/s0960129509990065","type":"journal-article","created":{"date-parts":[[2009,9,8]],"date-time":"2009-09-08T09:17:37Z","timestamp":1252401457000},"page":"943-957","source":"Crossref","is-referenced-by-count":5,"title":["The sequential topology on  is not regular"],"prefix":"10.1017","volume":"19","author":[{"given":"MATTHIAS","family":"SCHR\u00d6DER","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2009,9,8]]},"reference":[{"key":"S0960129509990065_ref16","volume-title":"General Topology","author":"Willard","year":"1970"},{"key":"S0960129509990065_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-56999-9"},{"key":"S0960129509990065_ref14","volume-title":"Proceedings of thirteenth ALGI Workshop","author":"Simpson","year":"2003"},{"key":"S0960129509990065_ref13","unstructured":"Schr\u00f6der M. 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