{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T21:57:31Z","timestamp":1747173451114,"version":"3.40.5"},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"8","license":[{"start":{"date-parts":[[2023,2,5]],"date-time":"2023-02-05T00:00:00Z","timestamp":1675555200000},"content-version":"unspecified","delay-in-days":157,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2022,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000014_inline1.png\"\/><jats:tex-math>\n$T_{0}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> spaces <jats:italic>X<\/jats:italic> and <jats:italic>Y<\/jats:italic>, it is proved that the following three conditions are equivalent: (1) the Scott space <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000014_inline2.png\"\/><jats:tex-math>\n$\\Sigma \\mathcal O(X)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of the lattice of all open sets of <jats:italic>X<\/jats:italic> is H-sober; (2) for every H-sober space <jats:italic>Y<\/jats:italic>, the function space <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000014_inline3.png\"\/><jats:tex-math>\n$\\mathbb{C}(X, Y)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of all continuous mappings from <jats:italic>X<\/jats:italic> to <jats:italic>Y<\/jats:italic> equipped with the Isbell topology is H-sober; (3) for every H-sober space <jats:italic>Y<\/jats:italic>, the Isbell topology on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000014_inline4.png\"\/><jats:tex-math>\n$\\mathbb{C}(X, Y)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> has property S with respect to H. One immediate corollary is that for a <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000014_inline5.png\"\/><jats:tex-math>\n$T_{0}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> space <jats:italic>X<\/jats:italic>, <jats:italic>Y<\/jats:italic> is a <jats:italic>d<\/jats:italic>-space (resp., well-filtered space) iff the function space <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000014_inline6.png\"\/><jats:tex-math>\n$\\mathbb{C}(X, Y)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> equipped with the Isbell topology is a <jats:italic>d<\/jats:italic>-space (resp., well-filtered space). It is shown that for any <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000014_inline7.png\"\/><jats:tex-math>\n$T_0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> space <jats:italic>X<\/jats:italic> for which the Scott space <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000014_inline8.png\"\/><jats:tex-math>\n$\\Sigma \\mathcal O(X)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is non-sober, the function space <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000014_inline9.png\"\/><jats:tex-math>\n$\\mathbb{C}(X, \\Sigma 2)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> equipped with the Isbell topology is not sober. The function spaces <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000014_inline10.png\"\/><jats:tex-math>\n$\\mathbb{C}(X, Y)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> equipped with the Scott topology, the compact-open topology and the pointwise convergence topology are also discussed. Our study also leads to a number of questions, whose answers will deepen our understanding of the function spaces related to H-sober spaces.<\/jats:p>","DOI":"10.1017\/s0960129523000014","type":"journal-article","created":{"date-parts":[[2023,2,5]],"date-time":"2023-02-05T23:07:00Z","timestamp":1675638420000},"page":"1099-1116","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["On function spaces equipped with Isbell topology and Scott topology"],"prefix":"10.1017","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1159-8477","authenticated-orcid":false,"given":"Xiaoquan","family":"Xu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Meng","family":"Bao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8976-5701","authenticated-orcid":false,"given":"Xiaoyuan","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2023,2,5]]},"reference":[{"key":"S0960129523000014_ref19","doi-asserted-by":"publisher","DOI":"10.4064\/fm704-4-2020"},{"key":"S0960129523000014_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2008.06.019"},{"key":"S0960129523000014_ref7","article-title":"Cartesian Closed Categories of Domains","volume":"66","author":"Jung","year":"1989","journal-title":"CWI Tract"},{"key":"S0960129523000014_ref2","first-page":"999","article-title":"On function spaces","volume":"17","author":"Ershov","year":"2020","journal-title":"Siberian Electronic Mathematical Reports"},{"key":"S0960129523000014_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2017.08.049"},{"key":"S0960129523000014_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2020.107323"},{"key":"S0960129523000014_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511542725"},{"key":"S0960129523000014_ref13","unstructured":"Scott, D. (1970). Outline of a mathematical theory of computation. In: Proceedings of the 4th Annual Princeton Conference on Information Sciences and Systems, Princeton University Press, 169\u2013176."},{"key":"S0960129523000014_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2019.106869"},{"key":"S0960129523000014_ref16","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2017.06.002"},{"key":"S0960129523000014_ref20","doi-asserted-by":"publisher","DOI":"10.1016\/j.jlamp.2014.10.003"},{"key":"S0960129523000014_ref12","doi-asserted-by":"publisher","DOI":"10.1137\/0205035"},{"volume-title":"Warzawa","year":"1989","author":"Engelking","key":"S0960129523000014_ref1"},{"key":"S0960129523000014_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2021.107757"},{"key":"S0960129523000014_ref8","doi-asserted-by":"publisher","DOI":"10.1109\/LICS.1990.113731"},{"key":"S0960129523000014_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(83)90095-6"},{"key":"S0960129523000014_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0089911"},{"key":"S0960129523000014_ref4","volume-title":"New Mathematical Monographs","volume":"22","author":"Goubault-Larrecq","year":"2003"},{"key":"S0960129523000014_ref5","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1982-0656096-4"},{"key":"S0960129523000014_ref17","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2020.107548"}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129523000014","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,5]],"date-time":"2023-04-05T02:00:07Z","timestamp":1680660007000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129523000014\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9]]},"references-count":20,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2022,9]]}},"alternative-id":["S0960129523000014"],"URL":"https:\/\/doi.org\/10.1017\/s0960129523000014","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"type":"print","value":"0960-1295"},{"type":"electronic","value":"1469-8072"}],"subject":[],"published":{"date-parts":[[2022,9]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}