Beside the Vietoris topology (thoroughly studied by Kuratowski, Michael, Choban  and more recently e.g. by Keesling, McCoy, Mizokami, Holá, Bouziad ), a considerable attention has been given to another hit-and-miss topology, the so-called Fell topology (note the work of e.g Mrowka, McCoy, Beer ). A remarkable property of this topology is that it is always compact. This makes the Fell topology useful in various applications, in particular, in the study of lower semicontinuous functions, random semicontinuous functions and random capacities by probabilists; it plays an important role in convergence and minimization problems for lower semicontinuous functions.

Beside the Hausdorff metric topology, the best known and probably the most important weak hyperspace topology is the Wijsman topology, introduced in the context of convex analysis and then studied in abstract by a number of authors (e.g. Levi, Lechicki, Hess, Beer, Costantini ). The interest for weak topologies arises among others from the fact that under some natural conditions they are measurably compatible, which in turn allows to express the multifunction measurability as ordinary measurability of an associated single-valued function. The general program aimed at characterization of topologies as weak topologies was carried out by Beer and his associates over the last ten years.

There has been an impressive growth in the number of recently introduced hyperspace topologies owing to their usefulness in various applications (such as probability, statistics or variational problems, for instance). It also explains the effort in understanding their structure, common features and general patterns, in order to find a common description for them. This general study of hyperspaces was undertaken in the seminal papers of Poppe  and more recently by Beer, Holá, Di Maio, Levi; moreover, papers of Sonntag, Zalinescu, Lucchetti, Pasquale  are partially or completely devoted to this goal, offering various levels of generalization.

Summary of my results

1. Topological games and hyperspaces. The Banach-Mazur game was first employed by McCoy  to obtain results on Baireness of the Vietoris topology; my results in [1] and [5] expanded on this technique and yielded applicable sufficient conditions for Baireness of general (proximal) hit-and-miss and various weak hypertopologies. In particular, I showed that a completely metrizable space gives rise to a Baire Wijsman topology. This completes an interesting line of research in a satisfactory manner:

• Effros  provided a Polish space having a Polish Wijsman topology;
• Beer   showed much later that the Wijsman topology generated by a complete separable metric space is Polish;
• Costantini   demonstrated that any compatible Polish metric generates a Polish Wijsman topology, but a complete metric space may have a non-Čech-complete Wijsman hyperspace;
• Pol  and Chaber,   answering a question of mine, showed that a complete metric space may even have a non-hereditarily Baire Wijsman hyperspace.

2. Common description of hypertopologies. As I mentioned earlier, there has been a considerable effort to find a common description for hypertopologies with various levels of generalization. In [4], I proposed a description based on the notion of approach spaces introduced by Lowen, which incorporates all the (proximal) hit-and-miss topologies as well as a big portion of weak hypertopologies; I also characterized the separation axioms up to complete regularity as well as metrizability and complete metrizability in this general setting; thus generalizing previous results on special hypertopologies spread all over the literature.

3. Hypertopologies with non-T₁ base spaces. Ever since Michael  introduced the notion of an admissible hypertopology, requiring thus the base space X to be T₁, it was generally assumed that the general treatment of hyperspace topologies could be rather complicated if non-T₂, let alone non-T₁, base spaces were considered. Beer writes: "Although the work of Vervaat, O'Brien and Norberg on random semicontinuous functions and capacities shows that one is often forced to work in non-Hausdorff spaces, it is considerably more difficult to investigate the connection between topological properties of the Fell topology and those of the underlying space at this level of generality."   My papers [1], [2], [6] and [4] show that for various important properties (such as metrizability, separation and countability axioms, uniformization or complete metrizability) of even general hypertopologies these difficulties do not occur.

4. Partial maps with closed domains. The so-called generalized compact-open topology on the space of partial maps with domains that are closed in a topological space X has been studied in connection with problems arising in differential equations, in mathematical economics, in convergence of dynamic programming models and other fields. The generalized compact-open topology t is completely metrizable, provided X  is hemicompact metrizable. I have shown [8] (independently with Ľ. Holá), that t is a Baire space if X  is locally compact and paracompact; further, that t  is Čech-complete if and only if X  is locally compact and hemicompact. Other properties of t, as well as its relationship to the compact-open topology of function spaces and the Fell hyperspace topology is also investigated in [8] , [9].

5. Baire spaces.

References