Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps

Some notes on densely continuous forms

“…Now, if (K (R), H) denotes the space of all nonempty compact subsets of the reals equipped with the Hausdorff metric, then F p X, K (R) can be seen as a subspace of F p X, 2 R . Moreover, we know that every form in D * (X) assumes only nonempty compact values (see [HV2,Lemma 2…”

“…φ is upper semicontinuous and has nonempty compact values at the points of U ; see B e e r [Be] or [HV1], [HV2] for definitions). Choose …”

“…Note 4.1º The collection of all results of [HV1] together with [HV2] and some previous results concerning function-space topologies can be found in [Va].…”

“…They also have an important relationship with minimal USCO maps (see D r e w n o w s k i and L a b u d a [DL]). For example, it has been shown in [HV2] (and earlier in [Ho1] for the real-valued case) that if X is Baire and Y is locally compact metric, then D * (X, Y ) is the space of all minimal USCO maps from X to Y . It is well known that the minimal USCO maps are a very convenient tool in the theory of games (see C h r i s t e n s e n [Ch] or S a i n t -R a y m o n d [Sa]) or in functional analysis (see the paper of B o r w e i n and M o o r s [BM] where a differentiability property of single-valued functions is characterized by their Clarke subdifferentials being convex minimal USCO maps).…”

“…Other properties of DC forms might also be of interest, such as the cardinal functions of topologies on such spaces (see H o lá and M c C o y [HoM] or [HV1]) or the selection properties of DC forms (see H o lá and H o lý [HH]) or relations of various topologies on spaces of DC forms (see [HV2] or H o lý [Ho]). …”