{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T13:13:59Z","timestamp":1700572439456},"reference-count":7,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2010,5,19]],"date-time":"2010-05-19T00:00:00Z","timestamp":1274227200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2010,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>It is well known that, in a topological space, the open sets can be characterized using ?lter convergence. In ZF (<jats:italic>Zermelo\u2010Fraenkel set theory without the Axiom of Choice<\/jats:italic>), we cannot replace filters by ultrafilters. It is proven that the ultra?lter convergence determines the open sets for every topological space if and only if the <jats:italic>Ultrafilter Theorem<\/jats:italic> holds. More, we can also prove that the Ultra?lter Theorem is equivalent to the fact that <jats:italic>u<jats:sub>X<\/jats:sub><\/jats:italic> = <jats:italic>k<jats:sub>X<\/jats:sub><\/jats:italic> for every topological space <jats:italic>X<\/jats:italic>, where <jats:italic>k<\/jats:italic> is the usual Kuratowski closure operator and <jats:italic>u<\/jats:italic> is the Ultra?lter Closure with <jats:italic>u<jats:sub>X<\/jats:sub><\/jats:italic> (<jats:italic>A<\/jats:italic>):= {<jats:italic>x<\/jats:italic> \u2208 <jats:italic>X<\/jats:italic>: (\u2203 <jats:italic>U<\/jats:italic> ultrafilter in <jats:italic>X<\/jats:italic>)[<jats:italic>U<\/jats:italic> converges to <jats:italic>x<\/jats:italic> and <jats:italic>A<\/jats:italic> \u2208 <jats:italic>U<\/jats:italic> ]}. However, it is possible to built a topological space <jats:italic>X<\/jats:italic> for which <jats:italic>u<jats:sub>X<\/jats:sub><\/jats:italic> \u2260 <jats:italic>k<jats:sub>X<\/jats:sub><\/jats:italic>, but the open sets are characterized by the ultra?lter convergence. To do so, it is proved that if every set has a free ultra?lter, then the Axiom of Countable Choice holds for families of non\u2010empty finite sets. It is also investigated under which set theoretic conditions the equality <jats:italic>u<\/jats:italic> = <jats:italic>k<\/jats:italic> is true in some subclasses of topological spaces, such as metric spaces, second countable T0\u2010spaces or {\u211d} (\u00a9 2010 WILEY\u2010VCH Verlag GmbH &amp; Co. KGaA, Weinheim)<\/jats:p>","DOI":"10.1002\/malq.200910014","type":"journal-article","created":{"date-parts":[[2010,5,20]],"date-time":"2010-05-20T08:24:42Z","timestamp":1274343882000},"page":"331-336","source":"Crossref","is-referenced-by-count":0,"title":["The Ultrafilter Closure in ZF"],"prefix":"10.1002","volume":"56","author":[{"given":"Gon\u00e7alo","family":"Gutierres","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2010,5,19]]},"reference":[{"key":"e_1_2_1_2_2","first-page":"329","article-title":"A model without ultrafilters, Bull","volume":"25","author":"Blass A.","year":"1977","journal-title":"Acad. Polon. Sci., S\u00e9r. Sci. Math Astron. Phys."},{"key":"e_1_2_1_3_2","unstructured":"G.Gutierres O Axioma da Escolha Numer\u00e1vel em Topologia. Ph. D. thesis Universidade de Coimbra 2004."},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200710018"},{"key":"e_1_2_1_5_2","unstructured":"H.Herrlich Axiom of Choice. Lecture Notes in Mathematics vol. 1876 (Springer\u2010Verlag 2006)."},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-8641(99)00132-7"},{"key":"e_1_2_1_7_2","doi-asserted-by":"crossref","unstructured":"P.Howard andJ. E.Rubin Consequences of the Axiom of Choice. Mathematical Surveys and Monographs vol. 59 (American Mathematical Society 1998).","DOI":"10.1090\/surv\/059"},{"key":"e_1_2_1_8_2","unstructured":"T. J.Jech The Axiom of Choice. Studies in Logic and the Foundations of Mathematics vol. 75 (North\u2010Holland Publ. Co. 1973)."}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.200910014","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.200910014","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,21]],"date-time":"2023-11-21T12:56:26Z","timestamp":1700571386000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.200910014"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,5,19]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2010,6]]}},"alternative-id":["10.1002\/malq.200910014"],"URL":"https:\/\/doi.org\/10.1002\/malq.200910014","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,5,19]]}}}