Dear Claudio, For locales, the questions about bundles can be reduced to results about spaces (= locales) internal in the topos of sheaves over the base space. By "bundle" I just mean "map" (always assumed continuous), but it is useful to have the alternative word because we take an alternative perspective in which the bundle is thought of as a space (the fibre) continuously parametrized by a base point. Then there are correspondences between properties of the bundle as map and its properties "fibrewise", the latter corresponding to properties of the internal space. local homeomorphism <-> fibrewise discrete perfect <-> fibrewise compact regular (or completely regular? They become inequivalent in a topos.) Now the localic version of your questions can be reduced to the topos-validity of questions about exponentiation of locales. If X compact regular and Y discrete, (1) Is Y^X discrete? (2) Is X^Y compact regular? I'm reasonably sure (2) is known, but I can't think of the references offhand. The hard part is Tychonoff, which is topos-valid localically. As for (1), I'm inclined to believe it but can't think of reasons or references at the moment. The "localic bundle theorem" that relates bundles over a topos to locales in it is fundamental and goes back - I believe - to Fourman, Scott, Joyal, Tierney. It is a result that depends on using point-free rather than point-set topology. All the same, I guess there's a chance of exploiting it to prove results about classical point-set bundles. "Topos-valid locale" sounds scary - Do we have to use lattices instead of spaces? Do we have to know all about toposes? A lot of my work has been about the beneficial effect of using the "geometric" fragment of topos-valid logic, using which one can reason with the points of a point-free space. All the best, Steve Vickers. On Mon, 15 Aug 2011 20:03:22 +0100 (BST), claudio pisani <pisclau@yahoo.it> wrote:
Dear Steve, by "perfect" I mean proper and separated (proper = closed with compact fibers = stably closed). I prefer the topological setting because I am not very acquainted with locales, but any result which seems to confirm the conjecture is good to me. So you are saying that (apart from the separation condition) the second question has an affirmative answer in Loc. Any idea about the first one?
Regards, Claudio
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