{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T16:43:49Z","timestamp":1767199429688,"version":"build-2238731810"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":9232,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1988,12]]},"abstract":"<jats:p>\n                    Let\n                    <jats:italic>M<\/jats:italic>\n                    be an o-minimal structure or a\n                    <jats:italic>p<\/jats:italic>\n                    -adically closed field. Let\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200027985_inline1.png\"\/>\n                    be the space of complete\n                    <jats:italic>n<\/jats:italic>\n                    -types over\n                    <jats:italic>M<\/jats:italic>\n                    equipped with the following topology: The basic open sets of\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200027985_inline1.png\"\/>\n                    are of the form \u0168 = {\n                    <jats:italic>p<\/jats:italic>\n                    \u2208\n                    <jats:italic>\n                      S\n                      <jats:sub>n<\/jats:sub>\n                    <\/jats:italic>\n                    (\n                    <jats:italic>M<\/jats:italic>\n                    ):\n                    <jats:italic>U<\/jats:italic>\n                    \u2208\n                    <jats:italic>p<\/jats:italic>\n                    } for\n                    <jats:italic>U<\/jats:italic>\n                    an\n                    <jats:italic>open<\/jats:italic>\n                    definable subset of\n                    <jats:italic>\n                      M\n                      <jats:sup>n<\/jats:sup>\n                    <\/jats:italic>\n                    .\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200027985_inline1.png\"\/>\n                    is a spectral space. (For\n                    <jats:italic>M<\/jats:italic>\n                    =\n                    <jats:italic>K<\/jats:italic>\n                    a real closed field,\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200027985_inline1.png\"\/>\n                    is precisely the\n                    <jats:italic>real spectrum<\/jats:italic>\n                    of\n                    <jats:italic>K<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    , \u2026,\n                    <jats:italic>\n                      X\n                      <jats:sub>n<\/jats:sub>\n                    <\/jats:italic>\n                    ]; see [CR].) We will equip\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200027985_inline1.png\"\/>\n                    with a sheaf of\n                    <jats:italic>\n                      L\n                      <jats:sub>M<\/jats:sub>\n                    <\/jats:italic>\n                    -structures (where\n                    <jats:italic>\n                      L\n                      <jats:sub>M<\/jats:sub>\n                    <\/jats:italic>\n                    is a suitable language). Again for\n                    <jats:italic>M<\/jats:italic>\n                    a real closed field this corresponds to the structure sheaf on\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200027985_inline1.png\"\/>\n                    (see [S]). Our main point is that when Th(\n                    <jats:italic>M<\/jats:italic>\n                    ) has definable Skolem functions, then if\n                    <jats:italic>p<\/jats:italic>\n                    \u2208\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200027985_inline1.png\"\/>\n                    , it follows that\n                    <jats:italic>M<\/jats:italic>\n                    (\n                    <jats:italic>p<\/jats:italic>\n                    ), the definable ultrapower of\n                    <jats:italic>M<\/jats:italic>\n                    at\n                    <jats:italic>p<\/jats:italic>\n                    , can be factored through\n                    <jats:italic>\n                      M\n                      <jats:sub>p<\/jats:sub>\n                    <\/jats:italic>\n                    , the stalk at\n                    <jats:italic>p<\/jats:italic>\n                    with respect to the above sheaf. This depends on the observation that if\n                    <jats:italic>M<\/jats:italic>\n                    \u227a\n                    <jats:italic>N, a<\/jats:italic>\n                    \u2208\n                    <jats:italic>\n                      N\n                      <jats:sup>n<\/jats:sup>\n                    <\/jats:italic>\n                    and\n                    <jats:italic>f<\/jats:italic>\n                    is an\n                    <jats:italic>M<\/jats:italic>\n                    -definable (partial) function defined at\n                    <jats:italic>a<\/jats:italic>\n                    , then there is an open\n                    <jats:italic>M<\/jats:italic>\n                    -definable set\n                    <jats:italic>U<\/jats:italic>\n                    \u2282\n                    <jats:italic>\n                      N\n                      <jats:sup>n<\/jats:sup>\n                    <\/jats:italic>\n                    with\n                    <jats:italic>a<\/jats:italic>\n                    \u2208\n                    <jats:italic>U<\/jats:italic>\n                    , and a\n                    <jats:italic>continuous M<\/jats:italic>\n                    -definable function\n                    <jats:italic>g<\/jats:italic>\n                    :\n                    <jats:italic>U<\/jats:italic>\n                    \u2192\n                    <jats:italic>N<\/jats:italic>\n                    such that\n                    <jats:italic>g(a)<\/jats:italic>\n                    =\n                    <jats:italic>f(a)<\/jats:italic>\n                    .\n                  <\/jats:p>\n                  <jats:p>\n                    In the case that\n                    <jats:italic>M<\/jats:italic>\n                    is an o-minimal expansion of a real closed field (or\n                    <jats:italic>M<\/jats:italic>\n                    is a\n                    <jats:italic>p<\/jats:italic>\n                    -adically closed field), it turns out that\n                    <jats:italic>M(p)<\/jats:italic>\n                    can be recovered as the unique quotient of\n                    <jats:italic>\n                      M\n                      <jats:sub>p<\/jats:sub>\n                    <\/jats:italic>\n                    which is an elementary extension of\n                    <jats:italic>M<\/jats:italic>\n                    .\n                  <\/jats:p>","DOI":"10.2307\/2274610","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:28:19Z","timestamp":1146940099000},"page":"1165-1169","source":"Crossref","is-referenced-by-count":9,"title":["Sheaves of continuous definable functions"],"prefix":"10.1017","volume":"53","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200027985","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T06:33:17Z","timestamp":1679466797000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200027985\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1988,12]]},"references-count":0,"aliases":["10.1017\/s0022481200027985"],"journal-issue":{"issue":"4","published-print":{"date-parts":[[1988,12]]}},"alternative-id":["S0022481200027985"],"URL":"https:\/\/doi.org\/10.2307\/2274610","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1988,12]]}}}