{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T20:04:42Z","timestamp":1744747482894},"reference-count":14,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2006,12]]},"abstract":"<jats:p> This paper gives a formalization of general topology in second-order arithmetic using countably based MF spaces. This formalization is used to study the reverse mathematics of general topology. <\/jats:p><jats:p> For each poset P we let MF (P) denote the set of maximal filters on P endowed with the topology generated by {N<jats:sub>p<\/jats:sub> | p \u2208 P}, where N<jats:sub>p<\/jats:sub> = {F \u2208 MF (P) | p \u2208 F}. We define a countably based MF space to be a space of the form MF (P) for some countable poset P. The class of countably based MF spaces includes all complete separable metric spaces as well as many nonmetrizable spaces. <\/jats:p><jats:p> The following reverse mathematics results are obtained. The proposition that every nonempty G<jats:sub>\u03b4<\/jats:sub> subset of a countably based MF space is homeomorphic to a countably based MF space is equivalent to [Formula: see text] over ACA<jats:sub>0<\/jats:sub>. The proposition that every uncountable closed subset of a countably based MF space contains a perfect set is equivalent over [Formula: see text] to the proposition that [Formula: see text] is countable for all A \u2286 \u2115. The proposition that every regular countably based MF space is homeomorphic to a complete separable metric space is equivalent to [Formula: see text] over [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0219061306000578","type":"journal-article","created":{"date-parts":[[2007,5,2]],"date-time":"2007-05-02T14:44:30Z","timestamp":1178117070000},"page":"203-232","source":"Crossref","is-referenced-by-count":15,"title":["REVERSE MATHEMATICS OF MF SPACES"],"prefix":"10.1142","volume":"06","author":[{"given":"CARL","family":"MUMMERT","sequence":"first","affiliation":[{"name":"University of Michigan, Department of Mathematics, 2074 East Hall\/530 Church Street, Ann Arbor, MI 48109, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,21]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/106\/1057814"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1990-1057041-5"},{"key":"rf4","series-title":"Studies in Logic and the Foundations of Mathematics","volume-title":"Set Theory","author":"Jech T.","year":"2002"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4190-4"},{"key":"rf6","series-title":"Studies in Logic and the Foundations of Mathematics","volume-title":"Set Theory","volume":"102","author":"Kunen K.","year":"1980"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129597002363"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1168352662"},{"key":"rf9","volume-title":"Recursive Aspects of Descriptive Set Theory","author":"Mansfield R.","year":"1985"},{"key":"rf10","unstructured":"D. A.\u00a0Martin and A. S.\u00a0Kechris, Analytic Sets, ed. C. H.\u00a0Rogers (Academic Press, 1980)\u00a0pp. 403\u2013470."},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-8641(02)00147-5"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.2178\/bsl\/1130335208"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.2307\/421132"},{"key":"rf17","unstructured":"M.\u00a0Schr\u00f6der, Computability and Complexity in Analysis, Informatik-Berichte\u00a0235, eds. K.\u00a0Ko (FernUniversit\u00e4t Hagen, 1998)\u00a0pp. 63\u201380."},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-59971-2"}],"container-title":["Journal of Mathematical Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0219061306000578","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T04:28:51Z","timestamp":1565152131000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0219061306000578"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,12]]},"references-count":14,"journal-issue":{"issue":"02","published-online":{"date-parts":[[2011,11,21]]},"published-print":{"date-parts":[[2006,12]]}},"alternative-id":["10.1142\/S0219061306000578"],"URL":"https:\/\/doi.org\/10.1142\/s0219061306000578","relation":{},"ISSN":["0219-0613","1793-6691"],"issn-type":[{"value":"0219-0613","type":"print"},{"value":"1793-6691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,12]]}}}