{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T12:56:34Z","timestamp":1772369794259,"version":"3.50.1"},"reference-count":29,"publisher":"World Scientific Pub Co Pte Ltd","issue":"02","funder":[{"name":"European Research Council","award":["338821"],"award-info":[{"award-number":["338821"]}]},{"name":"ISF","award":["181\/16"],"award-info":[{"award-number":["181\/16"]}]},{"name":"ISF","award":["1382\/15"],"award-info":[{"award-number":["1382\/15"]}]},{"name":"ISF","award":["181\/16"],"award-info":[{"award-number":["181\/16"]}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"crossref","award":["EXC 2044-390685587"],"award-info":[{"award-number":["EXC 2044-390685587"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"crossref","award":["CRC 878"],"award-info":[{"award-number":["CRC 878"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2020,8]]},"abstract":"<jats:p> We initiate the study of definable [Formula: see text]-topologies and show that there is at most one such [Formula: see text]-topology on a [Formula: see text]-henselian NIP field. Equivalently, we show that if [Formula: see text] is a bi-valued NIP field with [Formula: see text] henselian (respectively, [Formula: see text]-henselian), then [Formula: see text] and [Formula: see text] are comparable (respectively, dependent). <\/jats:p><jats:p> As a consequence, Shelah\u2019s conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. <\/jats:p><jats:p> We conclude by showing that Shelah\u2019s conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is [Formula: see text]-henselian. <\/jats:p>","DOI":"10.1142\/s0219061320500087","type":"journal-article","created":{"date-parts":[[2019,9,25]],"date-time":"2019-09-25T11:00:02Z","timestamp":1569409202000},"page":"2050008","source":"Crossref","is-referenced-by-count":11,"title":["Definable V-topologies, Henselianity and NIP"],"prefix":"10.1142","volume":"20","author":[{"given":"Yatir","family":"Halevi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Ben Gurion University of the Negev, Be\u2019er Sehva, Israel"}]},{"given":"Assaf","family":"Hasson","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ben Gurion University of the Negev, Be\u2019er Sehva, Israel"}]},{"given":"Franziska","family":"Jahnke","sequence":"additional","affiliation":[{"name":"Westf\u00e4lische Wilhelms-Universit\u00e4t M\u00fcnster, Institut f\u00fcr Mathematische Logik und Grundlagenforschung, Einsteinstr 62, 48149 M\u00fcnster, Germany"}]}],"member":"219","published-online":{"date-parts":[[2019,11,15]]},"reference":[{"key":"S0219061320500087BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0060932"},{"key":"S0219061320500087BIB003","doi-asserted-by":"publisher","DOI":"10.2307\/2373065"},{"key":"S0219061320500087BIB005","doi-asserted-by":"publisher","DOI":"10.2307\/2275808"},{"key":"S0219061320500087BIB006","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-51718-6_5"},{"key":"S0219061320500087BIB007","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090164"},{"key":"S0219061320500087BIB008","series-title":"Springer Monographs in Mathematics","volume-title":"Valued Fields","author":"Engler A. 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