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Math. Logic"],"published-print":{"date-parts":[[2026,1]]},"abstract":"Abstract<\/jats:title>\n \n In this paper, we investigate the poset\n \n \n $$\\textbf{OF}(X)$$<\/jats:tex-math>\n \n \n OF<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of free open filters on a given space\n X<\/jats:italic>\n . In particular, we characterize spaces for which\n \n \n $$\\textbf{OF}(X)$$<\/jats:tex-math>\n \n \n OF<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a lattice. For each\n \n \n $$n\\in \\mathbb {N}$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n we construct a scattered space\n X<\/jats:italic>\n such that\n \n \n $$\\textbf{OF}(X)$$<\/jats:tex-math>\n \n \n OF<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is order isomorphic to the\n n<\/jats:italic>\n -element chain, which implies the affirmative answer to two questions of Mooney. Assuming CH we construct a scattered space\n X<\/jats:italic>\n such that\n \n \n $$\\textbf{OF}(X)$$<\/jats:tex-math>\n \n \n OF<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is order isomorphic to\n \n \n $$(\\omega +1,\\ge )$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n \u03c9<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \u2265<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . To prove the latter facts we introduce and investigate a new stratification of ultrafilters which depends on scattered subspaces of\n \n \n $$\\beta (\\kappa )$$<\/jats:tex-math>\n \n \n \u03b2<\/mml:mi>\n (<\/mml:mo>\n \u03ba<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Assuming the existence of\n n<\/jats:italic>\n measurable cardinals, for every\n \n \n $$m_0,\\ldots ,m_{n}\\in \\mathbb {N}$$<\/jats:tex-math>\n \n \n \n m<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n m<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n we construct a space\n X<\/jats:italic>\n such that\n \n \n $$\\textbf{OF}(X)$$<\/jats:tex-math>\n \n \n OF<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is order isomorphic to\n \n \n $$\\prod _{i=0}^nm_i$$<\/jats:tex-math>\n \n \n \n \u220f<\/mml:mo>\n \n i<\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msubsup>\n \n m<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Also, we show that the existence of a metric space possessing a free\n \n \n $$\\omega _1$$<\/jats:tex-math>\n \n \n \u03c9<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -complete closed,\n \n \n $$G_\\delta $$<\/jats:tex-math>\n \n \n G<\/mml:mi>\n \u03b4<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ,\n \n \n $$F_{\\sigma }$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n \u03c3<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n or Borel ultrafilter is equivalent to the existence of a measurable cardinal.\n <\/jats:p>","DOI":"10.1007\/s00153-025-00985-2","type":"journal-article","created":{"date-parts":[[2025,9,17]],"date-time":"2025-09-17T11:46:28Z","timestamp":1758109588000},"page":"41-70","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Open filters and measurable cardinals"],"prefix":"10.1007","volume":"65","author":[{"given":"Serhii","family":"Bardyla","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jaroslav","family":"\u0160upina","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lyubomyr","family":"Zdomskyy","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,9,17]]},"reference":[{"key":"985_CR1","doi-asserted-by":"crossref","first-page":"175","DOI":"10.4153\/CJM-1951-022-3","volume":"3","author":"RH Bing","year":"1951","unstructured":"Bing, R.H.: Metrization of topological spaces. 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