{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T20:40:02Z","timestamp":1723063202798},"reference-count":0,"publisher":"Wiley","issue":"69","license":[{"start":{"date-parts":[[2000,1,1]],"date-time":"2000-01-01T00:00:00Z","timestamp":946684800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001501","name":"University Grants Commission","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001501","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2004,1]]},"abstract":"<jats:p>The main aim of this paper is to provide a construction of the Banaschewski compactification of a zero\u2010dimensional Hausdorff topological space as a structure space of a ring of ordered\nfield\u2010valued continuous functions on the space, and thereby exhibit the independence of the construction from any\ncompleteness axiom for an ordered field. In the process of describing this construction we have generalized the classical versions of M. H. Stone\u2032s theorem, the Banach\u2010Stone theorem, and\nthe Gelfand\u2010Kolmogoroff theorem. The paper is concluded with a conjecture of a split in the class of all\nzero\u2010dimensional but not strongly zero\u2010dimensional Hausdorff topological spaces into three classes that are labeled by\ninequalities between three compactifications of <jats:italic>X<\/jats:italic>, namely, the\nStone\u2010\u010cech compactification <jats:italic>\u03b2<\/jats:italic><jats:italic>X<\/jats:italic>, the Banaschewski\ncompactification <jats:italic>\u03b2<\/jats:italic><jats:sub>0<\/jats:sub><jats:italic>X<\/jats:italic>, and the structure space\n<jats:italic>&amp;#x1D510;<\/jats:italic><jats:sub><jats:italic>X<\/jats:italic>,<jats:italic>F<\/jats:italic><\/jats:sub> of the lattice\u2010ordered commutative ring <jats:italic>\u212d<\/jats:italic>(<jats:italic>X<\/jats:italic>, <jats:italic>F<\/jats:italic>) of all continuous functions on <jats:italic>X<\/jats:italic> taking values in the ordered\nfield <jats:italic>F<\/jats:italic>, equipped with its order topology. Some open problems\nare also stated.<\/jats:p>","DOI":"10.1155\/s0161171204305168","type":"journal-article","created":{"date-parts":[[2004,12,23]],"date-time":"2004-12-23T07:44:21Z","timestamp":1103787861000},"page":"3799-3816","update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Constructing Banaschewski compactification without Dedekind completeness axiom"],"prefix":"10.1155","volume":"2004","author":[{"given":"S. K.","family":"Acharyya","sequence":"first","affiliation":[]},{"given":"K. C.","family":"Chattopadhyay","sequence":"additional","affiliation":[]},{"given":"Partha Pratim","family":"Ghosh","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2004,12,23]]},"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2004\/216815.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/S0161171204305168","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T19:45:37Z","timestamp":1723059937000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/S0161171204305168"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1]]},"references-count":0,"journal-issue":{"issue":"69","published-print":{"date-parts":[[2004,1]]}},"alternative-id":["10.1155\/S0161171204305168"],"URL":"https:\/\/doi.org\/10.1155\/s0161171204305168","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2004,1]]},"assertion":[{"value":"2003-05-14","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2004-12-23","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}