The Fine Topology

Abstract

The topology of a topological space is the class of open subsets of the space. If T₁ and T₂ are topologies on a space, T₁ is said to be finer than T₂ (and then T₂ is said to be coarser than T₁) if T₂ ⊂ T₁. For any family of extended real-valued functions on a space there is a coarsest topology making every member of the family continuous, namely, the intersection of all the topologies doing this. The fine topology of classical potential theory is defined as the coarsest topology on ℝ^N making continuous every superharmonic function on ℝ^N. It is easy to verify that the fine and Euclidean topologies coincide when N = 1 (see Chapter XIV for classical potential theory on ℝ), and we suppose from now on in this chapter that N > 1. Concepts relative to the fine topology will be distinguished by an “f”, for example, f lim sup, δ _f A. From now on any otherwise unqualified topological concept will refer to the Euclidean topology. Since the fine topology is defined intrinsically in terms of superharmonic functions, it is not surprising that this topology plays a fundamental role in classical potential theory.