{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T15:33:00Z","timestamp":1747150380856,"version":"3.40.5"},"reference-count":31,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T00:00:00Z","timestamp":1690156800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2023,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A Hausdorff space  is called effectively Hausdorff if there exists a function <jats:italic>F<\/jats:italic>\u2014called a Hausdorff operator\u2014such that, for every  with , , where <jats:italic>U<\/jats:italic> and <jats:italic>V<\/jats:italic> are disjoint open neighborhoods of <jats:italic>x<\/jats:italic> and <jats:italic>y<\/jats:italic>, respectively. Among other results, we establish the following in , i.e., in Zermelo\u2013Fraenkel set theory without the Axiom of Choice ():<\/jats:p><jats:p><jats:list list-type=\"explicit-label\">\n<jats:list-item><jats:p> is equivalent to \u201cFor every set <jats:italic>X<\/jats:italic>, the Cantor cube  is effectively Hausdorff\u201d. This enhances the result of Howard, Keremedis, Rubin and Rubin [13] that  is equivalent to \u201cHausdorff spaces are effectively Hausdorff\u201d in .<\/jats:p><\/jats:list-item>\n<jats:list-item><jats:p>The Boolean Prime Ideal Theorem and the statement \u201cFor every infinite set <jats:italic>X<\/jats:italic>, the Stone space  of the Boolean algebra  is effectively Hausdorff\u201d are mutually independent. In particular, the latter statement is not provable in .<\/jats:p><\/jats:list-item>\n<jats:list-item><jats:p>The Axiom of Choice for non\u2010empty subsets of  () is equivalent to each of \u201cSeparable Hausdorff spaces are effectively Hausdorff\u201d and \u201cThe Cantor cube  is effectively\u00a0Hausdorff\u201d.<\/jats:p><\/jats:list-item>\n<jats:list-item><jats:p>The Principle of Dependent Choices in conjunction with the Axiom of Choice for continuum sized families of non\u2010empty subsets of  does not imply the axiom of choice for partitions of . The latter independence result fills the gap in information in Howard and Rubin's book \u201cConsequences of the Axiom of\u00a0Choice\u201d.<\/jats:p><\/jats:list-item>\n<jats:list-item><jats:p>The axiom of countable choice for non\u2010empty subsets of  is equivalent to each of \u201cDenumerable Hausdorff spaces are effectively Hausdorff\u201d, \u201cDenumerable <jats:italic>T<\/jats:italic><jats:sub>3<\/jats:sub> spaces are completely normal\u201d and \u201cDenumerable Tychonoff spaces are Urysohn\u201d.<\/jats:p><\/jats:list-item>\n<\/jats:list>\n<\/jats:p>","DOI":"10.1002\/malq.202300004","type":"journal-article","created":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T12:12:45Z","timestamp":1690200765000},"page":"347-369","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Hausdorff operators in ZF$\\mathsf {ZF}$"],"prefix":"10.1002","volume":"69","author":[{"given":"Kyriakos","family":"Keremedis","sequence":"first","affiliation":[{"name":"Department of Mathematics University of the Aegean Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9114-3661","authenticated-orcid":false,"given":"Eleftherios","family":"Tachtsis","sequence":"additional","affiliation":[{"name":"Department of Statistics and Actuarial\u2010Financial Mathematics University of the Aegean Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2023,7,24]]},"reference":[{"key":"e_1_2_9_2_1","first-page":"329","article-title":"A model without ultrafilters","volume":"25","author":"Blass A.","year":"1977","journal-title":"Bull. 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