Local Compactness and Bases in various formulations of Topology www.paultaylor.eu/ASD/loccbv I would like to ask a specialist question before telling you about this draft paper. In the theory of continuous lattices there is a result due to Jimmie Lawson that says that, if a<<b but not a<<c then there is a Scott-open filter that contains b but not c. The idea is to iterate the interpolation property: a<<...<<b2<<b1<<b0=b. Of course this requires Dependent Choice. Does anyone have an example of a topos without Dependent Choice containing a locale without enough Scott-open filters like this? ----------------- I have posted a longer version of this announcement on "Constructive News", addressed to Formal Topologists: https://groups.google.com/forum/#!topic/constructivenews/i1YWESQPh0Y A basis for a locally compact space is given by a family of pairs of subspaces, one open and the other compact. This paper gives the complete axiomatisation of the relation on indices that says when a basic compact subspace is covered by a finite set of basic open ones. It is a completely rewritten version of one I circulated last year called "A Concise Presentation for Locally Compact Spaces" and in particular now also characterises continuous functions. This work (already in last year's version) is innovative in that the axiomatisation does not require the basis to be closed under finite unions and intersections, as had been done in previous work by Achim Jung and Philipp Sunderhauf and by me. Therefore the leading example of a basis - balls in a metric space - is included. This step shows a lot of the features of interval analysis, but generalised from R to locally compact spaces. An equivalence of categories is proved in Point-Set Topology, Locale Theory, Formal Topology and Abstract Stone Duality. It turns out that, even though one can describe the points, open sets and basic compact sets explicitly, in order to prove correctness in the classical case it is necessary to go via Locales and Formal Topology. Doing this in all four formulations of general topology and their associated logical foundations provides a setting in which to examine how they differ in logical strength. We can say precisely that a continuous function according to a particular definition is a certain relation between sets that are definable in the corresponding logical system. Paul Taylor [For admin and other information see: http://www.mta.ca/~cat-dist/ ]