Topology and measure in logics for region-based theories of space

The present paper explores the question of completeness of Lmincont and its extensions for individual topological spaces of interest: the real line, Cantor space, the rationals, and the infinite binary tree. A second aim is to study a different, algebraic model of logics for region-based theories of space, based on the Lebesgue measure algebra (or algebra of Borel subsets of the real line modulo sets of Lebesgue measure zero). As a model for point-free space, the algebra was first discussed in [2]. The main results of the paper are that Lmincont is weakly complete for any zero-dimensional, dense-in-itself metric space (including, e.g., Cantor space and the rationals); the extension Lmincont+(Con) is weakly complete for the real line and the Lebesgue measure contact algebra. We also prove that the logic Lmincont+(Univ) is weakly complete for the infinite binary tree.