{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T00:49:59Z","timestamp":1777423799932,"version":"3.51.4"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"4","license":[{"start":{"date-parts":[[2015,12,1]],"date-time":"2015-12-01T00:00:00Z","timestamp":1448928000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-sa\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2015,12,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>Using Mizar [9], and the formal topological space structure (<jats:monospace>FMT_Space_Str<\/jats:monospace>) [19], we introduce the three <jats:monospace>U-FMT<\/jats:monospace> conditions (<jats:monospace>U-FMT filter, U-FMT with point<\/jats:monospace> and <jats:monospace>U-FMT local<\/jats:monospace>) similar to those <jats:italic>V<jats:sub>I<\/jats:sub>\n                  <\/jats:italic>, <jats:italic>V<jats:sub>II<\/jats:sub>\n                  <\/jats:italic>, <jats:italic>V<jats:sub>III<\/jats:sub>\n                  <\/jats:italic> and <jats:italic>V<jats:sub>IV<\/jats:sub>\n                  <\/jats:italic> of the proposition 2 in [10]:\n<jats:disp-quote>\n                     <jats:p>If to each element <jats:italic>x<\/jats:italic> of a set <jats:italic>X<\/jats:italic> there corresponds a set <jats:italic>B<\/jats:italic>(<jats:italic>x<\/jats:italic>) of subsets of <jats:italic>X<\/jats:italic> such that the properties <jats:italic>V<jats:sub>I<\/jats:sub>\n                        <\/jats:italic>, <jats:italic>V<jats:sub>II<\/jats:sub>\n                        <\/jats:italic>, <jats:italic>V<jats:sub>III<\/jats:sub>\n                        <\/jats:italic> and <jats:italic>V<jats:sub>IV<\/jats:sub>\n                        <\/jats:italic> are satisfied, then there is a unique topological structure on <jats:italic>X<\/jats:italic> such that, for each <jats:italic>x \u2208 X<\/jats:italic>, <jats:italic>B<\/jats:italic>(<jats:italic>x<\/jats:italic>) is the set of neighborhoods of <jats:italic>x<\/jats:italic> in this topology.<\/jats:p>\n                  <\/jats:disp-quote>\n               <\/jats:p>\n               <jats:p>We present a correspondence between a topological space and a space defined with the formal topological space structure with the three <jats:monospace>U-FMT<\/jats:monospace> conditions called the topology from neighbourhoods. For the formalization, we were inspired by the works of Bourbaki [11] and Claude Wagschal [31].<\/jats:p>","DOI":"10.1515\/forma-2015-0023","type":"journal-article","created":{"date-parts":[[2016,3,30]],"date-time":"2016-03-30T23:40:48Z","timestamp":1459381248000},"page":"289-296","source":"Crossref","is-referenced-by-count":1,"title":["Topology from Neighbourhoods"],"prefix":"10.1515","volume":"23","author":[{"given":"Roland","family":"Coghetto","sequence":"first","affiliation":[{"name":"Rue de la Brasserie 5, 7100 La Louvi\u00e8re, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2016,3,25]]},"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/23\/4\/article-p289.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2015-0023","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,9]],"date-time":"2021-04-09T03:31:23Z","timestamp":1617939083000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2015-0023"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,12,1]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,3,25]]},"published-print":{"date-parts":[[2015,12,1]]}},"alternative-id":["10.1515\/forma-2015-0023"],"URL":"https:\/\/doi.org\/10.1515\/forma-2015-0023","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2015,12,1]]}}}