Does any one know the answers to these questions? 1. Is trigonometry valid in toposes? (I'll be astonished if it isn't.) 2. Does a polynomial over the complex field C have only finitely many roots? More precisely: 1. Over any topos with nno, let R be the locale of "formal reals", i.e. the classifier for the geometric theory of Dedekind sections. Do sin, cos, arctan, etc. : R -> R exist and satisfy the expected properties? Are there general results (e.g. on power series) that say Yes, of course they do? 2. Consider the space S of square roots of the generic complex number. Working over C, it is the locale corresponding to the squaring map s: C -> C, z |-> z^2. The fibre over w is the space of square roots of w. s is not a local homeomorphism, so S is not a discrete locale. Hence we can't say S is even a set, let alone a finite set in any of the known senses. I don't believe its discretization pt(S) is Kuratowski finite either. If I've calculated it correctly, it is S except for having an empty stalk over zero (oops!), and there is no neighbourhood of zero on which an enumeration can be given of all the elements of pt(S). On the other hand, S is a Stone locale - one can easily construct the sheaf of Boolean algebras that is its lattice of compact opens. That sheaf of Boolean algebras is not Kuratowski finite, nor even, it seems to me, a subsheaf of a Kuratowski finite sheaf. So is there any sense at all in which S is finite? Steve Vickers Department of Pure Maths Faculty of Maths and Computing The Open University ----------- Tel: 01908-653144 Fax: 01908-652140 Web: http://mcs.open.ac.uk/sjv22