Bases for spaces Has anyone come across this was of seeing locally compact spaces? We say that a system B_n of VECTORS in a VECTOR SPACE is a BASIS if any other vector U can be expressed as a sum of scalar multiples of basic vectors. Likewise, we say that a system B_n of OPEN SUBSETS of a TOPOLOGICAL SPACE is a BASIS if any other open set U can be expressed as a "sum" (disjunction) of basic opens. How do we find out which basis elements contribute to the sum, and by what scalar multiple? By applying the DUAL BASIS A_n to the given element U, ie A_n.U Then U = sum_n A_n.U * B_n where - "sum" means linear sum, disjunction or existential quantification, - "scalars" in the case of topology range over the Sierpinski space, - the dot denotes - inner product of a dual vector with a vector to yield a scalar, - that U is an element of the family A_n, or - lambda application, - `*' denotes multiplication by a scalar or a vector, or conjunction. In topology, each A_n is a Scott-open family of open subsets. Unfortunately, in general it need not be a filter. In the case where it is a filter, it corresponds by the Hofmann--Mislove theorem to a compact saturated subspace K_n. Then if A_n.U is true, B_n < K_n < U where < denotes non-strict subset inclusion. In this case K_n provides a basis of compact neighbourhoods. Anyway, if A_n.B_n is true then B_n is itself compact open, and more generally A_n.U implies B_n << U ("way below"). Thus, A_n.U is defined in terms of - the traditional definition of local compactness as the existence of a compact subspace between B_n and U - continuous lattices as B_n << U. As you might have guessed, I discovered this during my current work on domain theory in Abstract Stone Duality, and have lambda-terms for A_n and B_n. Indeed, if such a basis exists for a space X then X is a Sigma-split subspace of Sigma^N (in the sense of my recently-announced paper "Subspaces in Abstract Stone Duality"), where i: X -> Sigma^N by x |--> lambda n. B_n.x I: Sigma^X -> Sigma^2 N by U |--> lambda psi. some n. A_n.U * psi.n make Sigma^X a retract of Sigma^2 N. Conversely, given i and I,we define a basis indexed, not by numbers themselves, but by lists (k) of numbers, by B_k.x = all n in k. (i.x).n A_k.U = (I.U).(lambda n.n in k) Paul Taylor (no academic affiliation) ASD web page: http://www.dcs.qmul.ac.uk/~pt/ASD 30-Jan-2002 19:52:11 -0400,2568;000000000001-00000000