{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T02:50:34Z","timestamp":1648522234709},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7954,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1992,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In an \u03c9<jats:sub>1<\/jats:sub>-saturated nonstandard universe a cut is an initial segment of the hyperintegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut <jats:italic>U<\/jats:italic>, a corresponding <jats:italic>U<\/jats:italic>-topology on the hyperintegers by letting <jats:italic>O<\/jats:italic> be <jats:italic>U<\/jats:italic>-open if for any <jats:italic>x<\/jats:italic> \u03f5 <jats:italic>O<\/jats:italic> there is a <jats:italic>y<\/jats:italic> greater than all the elements in <jats:italic>U<\/jats:italic> such that the interval [<jats:italic>x<\/jats:italic> \u2212 <jats:italic>y<\/jats:italic>, <jats:italic>x<\/jats:italic> + <jats:italic>y<\/jats:italic>] \u2286 <jats:italic>O<\/jats:italic>. Let <jats:italic>U<\/jats:italic> be a cut in a hyperfinite time line <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022817_inline1\" \/>, which is a hyperfinite initial segment of the hyperintegers. The <jats:italic>U<\/jats:italic>-monad topology of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022817_inline1\" \/> is the quotient topology of the <jats:italic>U<\/jats:italic>-topological space <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022817_inline1\" \/> modulo <jats:italic>U<\/jats:italic>. In this paper we answer a question of Keisler and Leth about the <jats:italic>U<\/jats:italic>-monad topologies by showing that when <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022817_inline1\" \/> is \u03ba-saturated and has cardinality \u03ba,(1) if the coinitiality of <jats:italic>U<\/jats:italic><jats:sub>1<\/jats:sub>, is uncountable, then the <jats:italic>U<\/jats:italic><jats:sub>1<\/jats:sub>,-monad topology and the <jats:italic>U<\/jats:italic><jats:sub>2<\/jats:sub>-monad topology are homcomorphic iff both <jats:italic>U<\/jats:italic><jats:sub>1<\/jats:sub>, and <jats:italic>U<\/jats:italic><jats:sub>2<\/jats:sub> have the same coinitiality; and (2)<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022817_inline1\" \/> can produce exactly three different <jats:italic>U<\/jats:italic>-monad topologies (up to homeomorphism) for those <jats:italic>U<\/jats:italic>'s with countable coinitiality. As a corollary <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022817_inline1\" \/> can produce exactly four different <jats:italic>U<\/jats:italic>-monad topologies if the cardinality of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022817_inline1\" \/> is \u03c9<jats:sub>1<\/jats:sub>.<\/jats:p>","DOI":"10.2307\/2275288","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:44:45Z","timestamp":1146955485000},"page":"534-539","source":"Crossref","is-referenced-by-count":0,"title":["<i>U<\/i>-monad topologies of hyperfinite time lines"],"prefix":"10.1017","volume":"57","author":[{"given":"Renling","family":"Jin","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200022817_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(88)90048-6"},{"key":"S0022481200022817_ref004","first-page":"71","volume":"56","author":"Keisler","year":"1991","journal-title":"Meager sets on the hyperfinite time line"},{"key":"S0022481200022817_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-444-86580-9.50009-4"},{"key":"S0022481200022817_ref006","volume-title":"Foundations of infinitesimal stochastic analysis","author":"Stroyan","year":"1986"},{"key":"S0022481200022817_ref002","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200022817_ref003","first-page":"1167","volume":"54","author":"Keisler","year":"1989","journal-title":"Descriptive set theory over hyperfinite sets"},{"key":"S0022481200022817_ref005","first-page":"1022","volume":"55","author":"Miller","year":"1990","journal-title":"Set theoretic properties of Loeb measure"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200022817","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T21:45:01Z","timestamp":1558043101000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200022817\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,6]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1992,6]]}},"alternative-id":["S0022481200022817"],"URL":"https:\/\/doi.org\/10.2307\/2275288","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,6]]}}}