A New Approach to a Hyperspace Theory

This article proposes a new method for a unified description of hypertopologies on the closed subsets of an arbitrary topological space X. In the metric case, it represents a natural generalization of the description of hypertopologies as weak topologies induced by gap functionals. The tools are a family of sets of real valued functions on X and a family of subsets of X. This method allows us to recover most of the standard hypertopologies, and naturally leads us to the definition of new hypertopologies.