{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T09:44:30Z","timestamp":1773222270631,"version":"3.50.1"},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":9689,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1987,9]]},"abstract":"<jats:p>In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof.<\/jats:p><jats:p>Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The \u201cmaterial basis\u201d for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a \u201cnicely behaved\u201d unstable structure and the latter the archetypal stable structure. In this sense we try here to situate our work on <jats:italic>o<\/jats:italic>-minimal structures [PS] in a general topological context. Note, however, that the <jats:italic>p<\/jats:italic>-adic numbers, and structures definable therein, will also fit into our analysis.<\/jats:p><jats:p>In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is \u201cexplicitly definable\u201d in the sense of Flum and Ziegler [FZ]. In \u00a72 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In \u00a73 we introduce a \u201cdimension rank\u201d on (closed) definable sets. In \u00a74 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions.<\/jats:p>","DOI":"10.2307\/2274362","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:22:39Z","timestamp":1146954159000},"page":"763-778","source":"Crossref","is-referenced-by-count":30,"title":["First order topological structures and theories"],"prefix":"10.1017","volume":"52","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200029741_bib013","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1976-14104-3"},{"key":"S0022481200029741_bib003","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0097006"},{"key":"S0022481200029741_bib010","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19750210145"},{"key":"S0022481200029741_bib005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0833698-1"},{"key":"S0022481200029741_bib001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(79)90019-6"},{"key":"S0022481200029741_bib002","unstructured":"van den Dries, L. and Scrowcroft, P. , On the structure of semialgebraic sets over p-adic fields, preprint, 1985."},{"key":"S0022481200029741_bib006","doi-asserted-by":"publisher","DOI":"10.2307\/2274029"},{"key":"S0022481200029741_bib012","volume-title":"Classification theory and the number of non-isomorphic models","author":"Shelah","year":"1978"},{"key":"S0022481200029741_bib008","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19760220107"},{"key":"S0022481200029741_bib004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90006-2"},{"key":"S0022481200029741_bib007","first-page":"605","volume":"41","author":"Macintyre","year":"1976","journal-title":"On definable subsets of p-adic fields"},{"key":"S0022481200029741_bib009","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0833697-X"},{"key":"S0022481200029741_bib011","doi-asserted-by":"publisher","DOI":"10.4064\/fm-81-2-159-171"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200029741","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T10:37:44Z","timestamp":1679481464000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200029741\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1987,9]]},"references-count":13,"aliases":["10.1017\/s0022481200029741"],"journal-issue":{"issue":"3","published-print":{"date-parts":[[1987,9]]}},"alternative-id":["S0022481200029741"],"URL":"https:\/\/doi.org\/10.2307\/2274362","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1987,9]]}}}