Hypertopologies and Applications
Hypertopologies, i.e. (admissible) topologies on the set of the closed subsets of a topological space, are not a central issue in optimization, nonlinear analysis, and well-posed problems. However, it is not a coincidence that a great impulse to the study of these topologies came in the recent years from people more involved in the former topics (and other, occasionally), rather than in general point set topology. The key point is that, when studying approximation and/or perturbation methods in analysis one needs soon topologies on closed sets, well suited especially for spaces that are not even locally compact, as e.g. infinite dimensional linear spaces. And, of course, this is not only matter of analysis: hypertopologies have important applications for instance in probability, statistics, mathematical economics, game theory. The aim of this paper is to describe and motivate a recent approach to the study and classification of hypertopologies, that seems to be particularly useful to deal with them for general applications, and to present some results in optimization, showing the use of these topologies. I do not make the effort of presenting here too formally these topics. My main goal is to give an idea how to look at them in view of the applications that I know, focusing especially on optimization problems and well-posedness.
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