This is to invite your comments on Local Compactness and the Baire Category Theorem in Abstract Stone Duality Paul Taylor http://www.di.unito.it/~pt/ASD Abstract Stone Duality is a re-axiomatisation of general topology intended to make it recursive. By turning the idea of the Scott topology on its head, notions that involve directed (infinitary) joins are reformulated using functions of higher type instead. Here we consider compactness and the WAY BELOW relation << used for continuous lattices. A new characterisation of LOCAL COMPACTNESS is formulated in terms of an EFFECTIVE BASIS, ie one that comes with a DUAL BASIS. This is used to prove a form of BAIRE'S CATEGORY THEOREM, that, for any countable family of open dense subsets, the intersection is also dense. This paper expands on the analogy with bases for vector spaces that I mentioned on "categories" on 30 January. It was accepted for "Category Theory and Computer Science" on the enthusiasm of the referee, although the version that was submitted was very sketchy, but it has now been completely rewritten. The first two ASD papers, "Sober Spaces and Continuations" and "Subspaces in ASD" have now been accepted by the referees for TAC, though I am waiting to coordinate them with "Local Compactness". There is also Scott Domains in Abstract Stone Duality Identifying the need for Scott domains to be overt (open) objects in intuitionistic locale theory, we re-work Scott's informations systems construction from first principles, to obtain a cartesian closed full subcategory in abstract Stone duality. The necessary and sufficient condition for overtness is that the consistency predicate be decidable. We also construct the halting set: an open subspace of N that is not closed. These two papers are the first in ASD to exploit the axiom that enforces Scott continuity: the first four did not use it at all in their core development. Nevertheless, "Geometric and Higher Order Logic" developed some familiar notions of general topology, with (in the absence of the Scott continuity axiom) a precise lattice duality between SUBSPACES QUANTIFIERS SPACES SEPARATION open existential overt discrete equality and closed universal compact Hausdorff inequality. The Scott continuity axiom of course spoils the precise duality, but it is still quite strongly apparent in "Local Compactness". Indeed, by dropping the requirement in locale theory and Bourbakian topology for all joins, in favour of the monadic property (which essentially has the effect of demanding recursively indexed joins instead), the notion of OVERT space becomes clearer and more natural. The reason for the unfamiliarity of overt spaces is that the arbitrary joins in locale theory and Bourbakian topology amount to saying that all spaces are overt. Both papers use overt subspaces, which are as important as compact ones, and there is no doubt that we do need this new word in place of Joyal & Tierney's OPEN locales. Johnstone and Vickers have also written about overt/open locales, but I don't know of other work. Can anyone help me here? Paul Taylor <pt@di.unito.it> but still physically in London. 18-Jun-2002 16:16:33 -0300,2400;000000000001-00000000