{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:21:28Z","timestamp":1753881688103,"version":"3.41.2"},"reference-count":46,"publisher":"World Scientific Pub Co Pte Ltd","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2023,4]]},"abstract":"<jats:p> We generalize the notion of analytic\/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic\/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that the Gromov\u2013Hausdorff distance maintains the same complexity if it is defined on the class of all Polish metric spaces, spaces bounded from below, from above, and from both below and above. Then we show that [Formula: see text] is not reducible to equivalences induced by orbit pseudometrics, generalizing the seminal result of Kechris and Louveau. We answer in negative a question of Ben Yaacov, Doucha, Nies, and Tsankov on whether balls in the Gromov\u2013Hausdorff and Kadets distances are Borel. In appendix, we provide new methods using games showing that the distance-zero classes in certain pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov. There is a complementary paper of the authors where reductions between the most common pseudometrics from functional analysis and metric geometry are provided. <\/jats:p>","DOI":"10.1142\/s0219061322500143","type":"journal-article","created":{"date-parts":[[2022,6,17]],"date-time":"2022-06-17T09:18:28Z","timestamp":1655457508000},"source":"Crossref","is-referenced-by-count":2,"title":["Complexity of distances: Theory of generalized analytic equivalence relations"],"prefix":"10.1142","volume":"23","author":[{"given":"Marek","family":"C\u00fath","sequence":"first","affiliation":[{"name":"Charles University, Faculty of Mathematics and Physics, Department of Mathematical, Analysis, Sokolovsk\u00e1 83, 186 75 Prague 8, Czech Republic"}]},{"given":"Michal","family":"Doucha","sequence":"additional","affiliation":[{"name":"Institute of Mathematics of the Czech Academy of Sciences, \u017ditn\u00e1 25, 115 67 Prague 1, Czech Republic"}]},{"given":"Ond\u0159ej","family":"Kurka","sequence":"additional","affiliation":[{"name":"Institute of Mathematics of the Czech Academy of Sciences, \u017ditn\u00e1 25, 115 67 Prague 1, Czech Republic"}]}],"member":"219","published-online":{"date-parts":[[2022,9,27]]},"reference":[{"key":"S0219061322500143BIB001","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1215\/00294527-2017-0026","volume":"59","author":"\u00c1lvarez L\u00f3pez J. 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