{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T15:50:20Z","timestamp":1770738620931,"version":"3.49.0"},"reference-count":5,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8777,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,3]]},"abstract":"<jats:p>In the last years several research projects have been motivated by the problem of constructing the usual geometrical spaces by admitting \u201cregions\u201d and \u201cinclusion\u201d between regions as primitives and by defining the points as suitable sequences or classes of regions (for references see [2]).<\/jats:p><jats:p>In this paper we propose and examine a system of axioms for the pointless space theory in which \u201cregions\u201d, \u201cinclusion\u201d, \u201cdistance\u201d and \u201cdiameter\u201d are assumed as primitives and the concept of point is derived. Such a system extends a system proposed by K. Weihrauch and U. Schreiber in [5].<\/jats:p><jats:p>In the sequel <jats:bold>R<\/jats:bold> and <jats:bold>N<\/jats:bold> denote the set of real numbers and the set of natural numbers, and E is a Euclidean metric space. Moreover, if <jats:italic>X<\/jats:italic> is a subset of <jats:bold>R<\/jats:bold>, then \u22c1<jats:italic>X<\/jats:italic> is the least upper bound and \u22c0<jats:italic>X<\/jats:italic> the greatest lower bound of <jats:italic>X<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2274963","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:33:48Z","timestamp":1146954828000},"page":"207-219","source":"Crossref","is-referenced-by-count":19,"title":["Pointless metric spaces"],"prefix":"10.1017","volume":"55","author":[{"given":"Giangiacomo","family":"Gerla","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026529_ref001","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1985.11971718"},{"key":"S0022481200026529_ref002","volume-title":"Handbook of incidence geometry","author":"Gerla"},{"key":"S0022481200026529_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(81)90027-X"},{"key":"S0022481200026529_ref003","unstructured":"Gerla G. , Distances, diameters and verisimilitude of theories (unpublished)."},{"key":"S0022481200026529_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-5921-9"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200026529","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T21:52:37Z","timestamp":1558216357000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200026529\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,3]]},"references-count":5,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1990,3]]}},"alternative-id":["S0022481200026529"],"URL":"https:\/\/doi.org\/10.2307\/2274963","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,3]]}}}