Topological and logical explorations of Krull dimension
Guram Bezhanishvili, Nick Bezhanishvili, Joel Lucero-Bryan, Jan van Mill
Abstract:
Krull dimension measures the depth of the spectrum Spec(R) of a commutative ring R. Since Spec(R) is a spectral space, Krull dimension can be defined for spectral spaces. Utilizing Stone duality, it can also be defined for distributive lattices. For an arbitrary topological space, the notion of Krull dimension is less useful. Isbell remedied this by introducing the concept of graduated dimension. In this paper we propose an alternate concept, that of localic Krull dimension of a topological space, which has its roots in modal logic. This is done by investigating the concept of Krull dimension for closure algebras and Heyting algebras, which formalize the notions of powerset and open set algebras of topological spaces. We compare localic Krull dimension to other well-known dimension functions, and show that it can detect topological differences between topological spaces that Krull dimension is unable to detect. We also investigate applications of localic Krull dimension to modal logic. We prove that for a T1-space to have a finite localic Krull dimension can be described by an appropriate generalization of the well-known concept of a nodec space. These considerations yield topological completeness and incompleteness results in modal logic that we examine in detail.