Local Compactness and Bases
in various formulations of Topology

Abstract A basis for a locally compact space is a family of pairs of subspaces, one open and the other compact, where containment of the compact subspace indicates whether the open one contributes to the union expressing a general open subspace. This is captured abstractly by saying which finite sets of basic opens cover a basic compact subspace. We identify the complete axiomatisation of this “way below” relation without assuming that the system is closed under unions or intersections, so balls in a metric space provide an example. We show how to reconstruct a space from an abstract basis in point–set topology, locale theory, formal topology and abstract Stone duality. We also characterise continuous functions by means of relations called matrices that generalise the way-below relation. Hence a category defined using relations is weakly equivalent to that of locally compact spaces in each of these four formulations of topology, according to its appropriate logical foundations, namely set theory with Choice, the logic of an elementary topos, Martin-Löf type theory and an arithmetic universe respectively. Subsequent work will develop abstract bases towards computation. This paper is part of the Abstract Stone Duality programme to re-axiomatise general topology in a computable fashion.

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