Fine topology (potential theory)

In potential theory, the fine topology is a topology on a locally compact Hausdorff space that is finer than the original (Euclidean) topology. It is defined using potential-theoretic concepts, specifically by making thin sets rare, and it plays a crucial role in refining the notion of "almost everywhere" in potential theory.

Definition:

A set E is said to be thin at a point x if the reduced function of E relative to a suitable measure is discontinuous at x. The fine topology is the weakest topology on the space for which every set that is thin at all its points is open. Equivalently, a set U is finely open if for every x in U, the complement of U is thin at x.

Properties:

• Finer than the original topology: Every open set in the original topology is also finely open, but the converse is not generally true. Thus, the fine topology has more open sets.

• Non-metrizable: The fine topology is generally not metrizable.

• Quasi-Lindelöf: The fine topology is quasi-Lindelöf, meaning that every open cover has a countable subcollection that covers all but a set of zero capacity.

• Irregular points: The notion of thinness and the fine topology are closely related to the concept of irregular boundary points in Dirichlet problems. A boundary point is irregular if and only if the complement of the domain is thin at that point.

• Fine continuity: A function is finely continuous if it is continuous with respect to the fine topology. Finely continuous functions have important properties related to the behavior of functions almost everywhere in the sense of capacity.

• Applications: The fine topology is crucial in several areas of potential theory, including:

  • Characterizing removable singularities of potentials.
  • Studying the boundary behavior of solutions to partial differential equations (especially the Laplace equation and related equations).
  • Refining the notion of "almost everywhere" and defining capacities.
  • Analyzing the convergence of sequences of potentials.