Heine-Borel Does not Imply the Fan Theorem

Introduction. 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. We will consider four formal spaces, or locales, namely formal Cantor space C, formal Baire space B, the formal real line R, and the formal function space RR being the exponential in the category of locales (cf. [3]). The corresponding spaces of points will be denoted by pt(C), pt(B), pt(R) and pt(RR). 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 C has enough points is equivalent to the compactness of the space of points pt(C), and is traditionally known in intuitionistic analysis as the Fan Theorem (FT). Similarly, the assertion that B has enough points is equivalent to the principle of (monotone) Bar Induction (BI). The locale R has enough points iff its space of points pt(R) is locally compact, i.e. the unit interval pt[O, 1] c pt(R) is compact, which is of course known as the Heine-Borel Theorem (HB). The statement that RR has enough points, i.e. that there are "enough" continuous functions from R 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. As is well known, (BI)-(FT)-(HB). A possible way of explaining that (BI) implies (FT) is by observing that B is homeomorphic to the exponential CC, as has recently been pointed out by Hyland [3]. In the present context, therefore, the exponential RR is a natural object of study. Note that (BI) (EF) since RR is countably presented, and hence a continuous image of B. The principle (FT) holds in every spatial topos, but (BI) does not, so the implication (BI) = (FT) is not reversible (cf. [2]). In [1, ?4.11], it was asked whether (HB) implies (FT). We will show that this is not the case by proving that R has enough points in sheaves over the locale K(R2) …