{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T08:20:41Z","timestamp":1648887641466},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10876,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1984,6]]},"abstract":"<jats:p>This paper deals with locales and their spaces of points in intuitionistic analysis or, if you like, in (Grothendieck) toposes. One of the important aspects of the problem whether a certain locale has enough points is that it is directly related to the (constructive) completeness of a geometric theory. A useful exposition of this relationship may be found in [1], and we will assume that the reader is familiar with the general framework described in that paper.<\/jats:p><jats:p>We will consider four formal spaces, or locales, namely formal Cantor space <jats:italic>C<\/jats:italic>, formal Baire space <jats:italic>B<\/jats:italic>, the formal real line <jats:italic>R<\/jats:italic>, and the formal function space <jats:italic>R<jats:sup>R<\/jats:sup><\/jats:italic> being the exponential in the category of locales (cf. [3]). The corresponding spaces of points will be denoted by pt(<jats:italic>C<\/jats:italic>), pt(<jats:italic>B<\/jats:italic>), pt(<jats:italic>R<\/jats:italic>) and pt(<jats:italic>R<jats:sup>R<\/jats:sup><\/jats:italic>). Classically, these locales all have enough points, of course, but constructively or in sheaves this may fail in each case. Let us recall some facts from [1]: the assertion that <jats:italic>C<\/jats:italic> has enough points is equivalent to the compactness of the space of points pt(<jats:italic>C<\/jats:italic>), and is traditionally known in intuitionistic analysis as the <jats:italic>Fan Theorem<\/jats:italic> (FT). Similarly, the assertion that <jats:italic>B<\/jats:italic> has enough points is equivalent to the principle of (monotone) <jats:italic>Bar Induction<\/jats:italic> (BI). The locale <jats:italic>R<\/jats:italic> has enough points iff its space of points pt(<jats:italic>R<\/jats:italic>) is locally compact, i.e. the unit interval pt[0, 1] \u2282 pt(<jats:italic>R<\/jats:italic>) is compact, which is of course known as the <jats:italic>Heine-Borel Theorem<\/jats:italic> (HB). The statement that <jats:italic>R<jats:sup>R<\/jats:sup><\/jats:italic> has enough points, i.e. that there are \u201cenough\u201d continuous functions from <jats:italic>R<\/jats:italic> to itself, does not have a well-established name. We will refer to it (not very imaginatively, I admit) as the principle (EF) of Enough Functions.<\/jats:p>","DOI":"10.2307\/2274182","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:08:28Z","timestamp":1146953308000},"page":"514-519","source":"Crossref","is-referenced-by-count":5,"title":["Heine-Borel does not imply the Fan Theorem"],"prefix":"10.1017","volume":"49","author":[{"given":"Ieke","family":"Moerdijk","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200033557_ref001","first-page":"107","volume-title":"Proceedings of the Brouwer Centenary Conference","author":"Fourman","year":"1982"},{"key":"S0022481200033557_ref002","first-page":"280","volume-title":"Applications of Sheaves (Proceedings of the Research Symposium, Durham, 1977)","volume":"753","author":"Fourman","year":"1979"},{"key":"S0022481200033557_ref003","first-page":"264","volume-title":"Continuous Lattices (Proceedings of the Conference (Workshop IV), Bremen, 1979)","volume":"871","author":"Hyland","year":"1981"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200033557","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T20:48:31Z","timestamp":1558644511000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200033557\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1984,6]]},"references-count":3,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1984,6]]}},"alternative-id":["S0022481200033557"],"URL":"https:\/\/doi.org\/10.2307\/2274182","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1984,6]]}}}