{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:35:08Z","timestamp":1759638908392},"reference-count":40,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2020,1,1]],"date-time":"2020-01-01T00:00:00Z","timestamp":1577836800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,1,23]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We develop the basic constructive theory of embeddings of Bishop spaces in parallel to the basic classical theory of embeddings of topological spaces. The theory of Bishop spaces is a constructive approach to point-function topology and a natural constructive alternative to the classical theory of the rings of continuous functions. Our most significant result is the translation of the classical Urysohn extension theorem within the theory of Bishop spaces. The related theory of the zero sets of a Bishop topology is also included. We work within $\\textrm{BISH}^{\\ast }$, Bishop\u2019s informal system of constructive mathematics $\\textrm{BISH}$ equipped with inductive definitions with rules of countably many premises.<\/jats:p>","DOI":"10.1093\/logcom\/exaa015","type":"journal-article","created":{"date-parts":[[2020,1,9]],"date-time":"2020-01-09T04:20:01Z","timestamp":1578543601000},"page":"349-379","source":"Crossref","is-referenced-by-count":4,"title":["Embeddings of Bishop spaces"],"prefix":"10.1093","volume":"30","author":[{"given":"Iosif","family":"Petrakis","sequence":"first","affiliation":[{"name":"Mathematics Institute, Ludwig-Maximilians Universit\u00e4t M\u00fcnchen, Germany"}]}],"member":"286","published-online":{"date-parts":[[2020,2,17]]},"reference":[{"key":"2020040702451732500_ref1","volume-title":"Foundations of Constructive Analysis","author":"Bishop","year":"1967"},{"key":"2020040702451732500_ref2","first-page":"53","article-title":"Mathematics as a numerical language","volume-title":"Intuitionism and Proof Theory","author":"Bishop","year":"1970"},{"key":"2020040702451732500_ref3","article-title":"Constructive measure theory","volume":"116","author":"Bishop","year":"1972","journal-title":"Memoirs of the American Mathematical Society"},{"key":"2020040702451732500_ref4","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61667-9","volume-title":"Constructive Analysis","author":"Bishop","year":"1985"},{"key":"2020040702451732500_ref5","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511565663","volume-title":"Varieties of Constructive Mathematics","author":"Bridges","year":"1987"},{"key":"2020040702451732500_ref6","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1002\/malq.200610006","article-title":"A constructive treatment of Urysohn\u2019s lemma in an apartness space","volume":"52","author":"Bridges","year":"2006","journal-title":"Mathematical Logic Quarterly"},{"key":"2020040702451732500_ref7","volume-title":"Techniques of Constructive Analysis","author":"Bridges","year":"2006"},{"key":"2020040702451732500_ref8","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-22415-7","volume-title":"Apartness and Uniformity: A Constructive Development","author":"Bridges","year":"2011"},{"key":"2020040702451732500_ref9","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.apal.2011.06.018","article-title":"Reflections on function spaces","volume":"163","author":"Bridges","year":"2012","journal-title":"Annals of Pure and Applied Logic"},{"key":"2020040702451732500_ref10","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1007\/BF01209181","article-title":"Zur Begr\u00fcndung der intuitionistische mathematik III","volume":"96","author":"Brouwer","year":"1926","journal-title":"Mathematische Annalen"},{"key":"2020040702451732500_ref11","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0091894","volume-title":"Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretic Studies","author":"Buchholz","year":"1981"},{"key":"2020040702451732500_ref12","volume-title":"Topology","author":"Dugundji","year":"1989"},{"key":"2020040702451732500_ref13","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4615-7819-2","volume-title":"Rings of Continuous Functions","author":"Gillman","year":"1960"},{"key":"2020040702451732500_ref14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0003-4843(81)90010-3","article-title":"Concepts of general topology in constructive mathematics and in sheaves","volume":"20","author":"Grayson","year":"1981","journal-title":"Annals of Mathematical Logic"},{"key":"2020040702451732500_ref15","article-title":"Sur quelques points du calcul functionnel","volume":"22","author":"Fr\u00e9chet","year":"1906","journal-title":"Rendiconti del Circolo Matematico di Palermo"},{"key":"2020040702451732500_ref16","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1016\/j.apal.2006.01.004","article-title":"Quasi-apartness and neighborhood spaces","volume":"141","author":"Ishihara","year":"2006","journal-title":"Annals of Pure and Applied Logic"},{"key":"2020040702451732500_ref17","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.apal.2011.06.022","article-title":"Two subcategories of apartness spaces","volume":"163","author":"Ishihara","year":"2013","journal-title":"Annals of Pure and Applied Logic"},{"key":"2020040702451732500_ref18","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1016\/j.apal.2012.10.009","article-title":"Relating Bishop\u2019s function spaces to neighborhood spaces","volume":"164","author":"Ishihara","year":"2013","journal-title":"Annals of Pure and Applied Logic"},{"key":"2020040702451732500_ref19","volume-title":"Notes on Constructive Mathematics","author":"Martin-L\u00f6f","year":"1968"},{"key":"2020040702451732500_ref20","volume-title":"Intuitionistic Type Theory: Notes by Giovanni Sambin on a Series of Lectures Given in Padua, June 1980","author":"Martin-L\u00f6f","year":"1984"},{"key":"2020040702451732500_ref21","first-page":"541","article-title":"Brouwer Dummett, and the bar theorem","volume-title":"Atti del Congresso Nazionale di Logica, Montecatini Terme, 1\u20135 Ottobre 1979","author":"Martino","year":"1981"},{"key":"2020040702451732500_ref22","doi-asserted-by":"crossref","first-page":"347","DOI":"10.2307\/2272159","article-title":"Constructive set theory","volume":"40","author":"Myhill","year":"1975","journal-title":"The Journal of Symbolic Logic"},{"key":"2020040702451732500_ref23","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1002\/malq.200310054","article-title":"On the regular extension axiom and its variants","volume":"49","author":"Lubarsky","year":"2003","journal-title":"Mathematical Logic Quarterly"},{"key":"2020040702451732500_ref24","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/978-1-4020-8926-8_12","article-title":"From intuitionistic to point-free topology: on the foundations of homotopy theory","volume-title":"Logicism, Intuitionism, and Formalism","author":"Palmgren","year":"2009"},{"key":"2020040702451732500_ref25","first-page":"302","article-title":"Completely regular Bishop spaces","volume-title":"Evolving Computability, CiE 2015","author":"Petrakis","year":"2015"},{"key":"2020040702451732500_ref26","volume-title":"Constructive Topology of Bishop Spaces","author":"Petrakis","year":"2015"},{"key":"2020040702451732500_ref27","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/978-3-319-27683-0_21","article-title":"The Urysohn extension theorem for Bishop spaces","volume-title":"Symposium on Logical Foundations in Computer Science 2016","author":"Petrakis","year":"2016"},{"key":"2020040702451732500_ref28","first-page":"605","article-title":"A constructive function-theoretic approach to topological compactness","volume-title":"Proceedings of the 31st Annual ACM-IEEEE Symposium on Logic in Computer Science (LICS 2016), July 5\u20138, 2016, NYC, USA","author":"Petrakis","year":"2016"},{"key":"2020040702451732500_ref29","first-page":"1","article-title":"Constructive uniformities of pseudometrics and Bishop topologies","author":"Petrakis","year":"2019","journal-title":"Journal of Logic and Analysis"},{"key":"2020040702451732500_ref30","first-page":"240","article-title":"Borel and Baire sets in Bishop spaces","volume-title":"CiE 2019","author":"Petrakis","year":"2019"},{"key":"2020040702451732500_ref31","article-title":"Direct spectra of Bishop spaces and their limits","author":"Petrakis","year":"2019","journal-title":"Logical Methods in Computer Science"},{"key":"2020040702451732500_ref32","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/978-3-319-58741-7_33","article-title":"McShane\u2013Whitney pairs","volume-title":"Unveiling Dynamics and Complexity, CiE 2017","author":"Petrakis","year":"2017"},{"key":"2020040702451732500_ref33","article-title":"Dependent sums and dependent products in Bishop\u2019s set theory","volume-title":"TYPES 2018, LIPIcs","author":"Petrakis","year":"2019"},{"key":"2020040702451732500_ref34","article-title":"McShane\u2013Whitney extensions in constructive analysis","author":"Petrakis","journal-title":"Logical Methods in Computer Science"},{"key":"2020040702451732500_ref35","volume-title":"Families of Sets in Bishop Set Theory","author":"Petrakis","year":"2020"},{"key":"2020040702451732500_ref36","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/978-1-4613-0897-3_12","article-title":"Intuitionistic formal spaces\u2014a first communication","volume-title":"Mathematical Logic and its Applications","author":"Sambin","year":"1987"},{"key":"2020040702451732500_ref37","volume-title":"The Basic Picture: Structures for Constructive Topology","author":"Sambin","year":""},{"key":"2020040702451732500_ref38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1215\/S0012-7094-63-03001-1","article-title":"Quasi-topologies","volume":"30","author":"Spanier","year":"1963","journal-title":"Duke Mathematical Journal"},{"key":"2020040702451732500_ref39","volume-title":"Intuitionistic General Topology","author":"Troelstra","year":"1966"},{"key":"2020040702451732500_ref40","article-title":"Analysis in metric spaces","author":"Tuominen","year":"2014","journal-title":"Lecture Notes"}],"container-title":["Journal of Logic and Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/logcom\/article-pdf\/30\/1\/349\/33016392\/exaa015.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/logcom\/article-pdf\/30\/1\/349\/33016392\/exaa015.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,4,7]],"date-time":"2020-04-07T06:45:30Z","timestamp":1586241930000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/logcom\/article\/30\/1\/349\/5739316"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":40,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,2,17]]},"published-print":{"date-parts":[[2020,1,23]]}},"URL":"https:\/\/doi.org\/10.1093\/logcom\/exaa015","relation":{},"ISSN":["0955-792X","1465-363X"],"issn-type":[{"value":"0955-792X","type":"print"},{"value":"1465-363X","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2020,1]]},"published":{"date-parts":[[2020,1]]}}}