{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T13:57:43Z","timestamp":1648735063324},"reference-count":2,"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":3571,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,6]]},"abstract":"<jats:p>Herzog and Rothmaler gave the following purely topological characterization of stable theories. (See the exercises 11.3.4 \u2013 11.3.7 in [2]).<\/jats:p><jats:p><jats:italic>A complete theory T is stable iff for any model M and any extension M<\/jats:italic> \u2282 <jats:italic>B the restriction map S<\/jats:italic>(<jats:italic>B<\/jats:italic>) \u2192 <jats:italic>S<\/jats:italic>(<jats:italic>M<\/jats:italic>) <jats:italic>has a continuous section<\/jats:italic>.<\/jats:p><jats:p>In fact, if <jats:italic>T<\/jats:italic> is stable, taking the unique non-forking extension defines a continuous section of S(<jats:italic>B<\/jats:italic>) \u2192 S(<jats:italic>A<\/jats:italic>) for all subsets <jats:italic>A<\/jats:italic> of <jats:italic>B<\/jats:italic>, provided <jats:italic>A<\/jats:italic> is algebraically closed in <jats:italic>T<\/jats:italic><jats:sup>eq<\/jats:sup>. Herzog and Rothmaler asked, if, for stable <jats:italic>T<\/jats:italic>, there is a continuous section for <jats:italic>any<\/jats:italic> subset <jats:italic>A<\/jats:italic> of <jats:italic>B<\/jats:italic>. Or, equivalently, if for any <jats:italic>A<\/jats:italic>, S(acl<jats:sup>eq<\/jats:sup>(<jats:italic>A<\/jats:italic>)) \u2192 S(<jats:italic>A<\/jats:italic>) has a continuous section.<\/jats:p><jats:p>This is an interesting problem, also for unstable <jats:italic>T<\/jats:italic>. Is it true that for any <jats:italic>T<\/jats:italic> and any set of parameters <jats:italic>A<\/jats:italic> the restriction map S(acl(<jats:italic>A<\/jats:italic>)) \u2192 S(<jats:italic>A<\/jats:italic>) has a continuous section? We answer the question by the following two theorems.<\/jats:p><jats:p>Theorem 1. <jats:italic>Let A be a subset of a model of T. Assume that the Boolean algebra of acl(<jats:italic>A<\/jats:italic>)-definable formulas is generated by<\/jats:italic><\/jats:p><jats:p>\u2022 <jats:italic>some countable set of formulas<\/jats:italic>,<\/jats:p><jats:p>\u2022 <jats:italic>all A\u2013definable formulas<\/jats:italic>,<\/jats:p><jats:p>\u2022 <jats:italic>all formulas which are atomic over<\/jats:italic> acl(<jats:italic>A<\/jats:italic>).<\/jats:p><jats:p><jats:italic>Then<\/jats:italic> S(acl(<jats:italic>A<\/jats:italic>)) \u2192 S(<jats:italic>A<\/jats:italic>) <jats:italic>has a continuous section<\/jats:italic>.<\/jats:p><jats:p>The conditions of the theorems are satisfied if, for example, <jats:italic>L<\/jats:italic> and <jats:italic>A<\/jats:italic> are countable, or, if there are only countably many non-isolated types over acl(<jats:italic>A<\/jats:italic>).<\/jats:p><jats:p>Theorem 2. <jats:italic>There is a theory of Morley rank<\/jats:italic> 2 <jats:italic>and Morley degree<\/jats:italic> 1 <jats:italic>such that<\/jats:italic> S(acl(\u2205)) \u2192 S(\u2205) <jats:italic>has no continuous section<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1082418538","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:36:23Z","timestamp":1109799383000},"page":"478-481","source":"Crossref","is-referenced-by-count":0,"title":["On a question of Herzog and Rothmaler"],"prefix":"10.1017","volume":"69","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]},{"given":"Martin","family":"Ziegler","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200007854_ref001","volume-title":"Handbook of Boolean Algebras","volume":"3","author":"Monk","year":"1989"},{"key":"S0022481200007854_ref002","volume-title":"Introduction to Model Theory","author":"Rothmaler","year":"2000"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200007854","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T21:02:55Z","timestamp":1557176575000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200007854\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,6]]},"references-count":2,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2004,6]]}},"alternative-id":["S0022481200007854"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1082418538","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,6]]}}}