Multivalued function spaces and Atsuji spaces

[EN] In this paper we present two themes. The first one describes a transparent treatment of some of the recent results in graph topologies on multi-valued functions. The study includes Vietoris topology, Fell topology, Fell uniform topology on compacta and uniform topology on compacta. The second theme concerns when continuity is equivalent to proximal continuity or uniform continuityNaimpally, S. (2003). Multivalued function spaces and Atsuji spaces. Applied General Topology. 4(2):201-209. doi:10.4995/agt.2003.2025.SWORD2012094

All hypertopologies are hit-and-miss

[EN] We solve a long standing problem by showing that all known hypertopologies are hit-and-miss. Our solution is not merely of theoretical importance. This representation is useful in the study of comparison of the Hausdorff-Bourbaki or H-B uniform topologies and the Wijsman topologies among themselves and with others. Up to now some of these comparisons needed intricate manipulations. The H-B uniform topologies were the subject of intense activity in the 1960's in connection with the Isbell-Smith problem. We show that they are proximally locally finite topologies from which the solution to the above problem follows easily. It is known that the Wijsman topology on the hyperspace is the proximal ball (hit-and-miss) topology in”nice” metric spaces including the normed linear spaces. With the introduction of a new far-miss topology we show that the Wijsman topology is hit-and-miss for all metric spaces. From this follows a natural generalization of the Wijsman topology to the hyperspace of any T1 space. Several existing results in the literature are easy consequences of our workNaimpally, S. (2002). All hypertopologies are hit-and-miss. Applied General Topology. 3(1):45-53. doi:10.4995/agt.2002.2111SWORD45533

Proximal convergence

Suppose X is a topological space and Y a proximity space , fn L C f (Leader Convergence) iff for each A in X, B in Y, f(A) near B implies eventually fn (A) is near B. L.C. is a generalization of U. C. (Uniform Convergence). In this paper we study L. C. and various generalizations and prove analogues of the classical results of Arzelà, Dini and others

Strong Proximal Continuity and Convergence

In several situations the notion of uniform continuity can be strengthened to strong uniform continuity to produce interesting properties, especially in constrained problems. The same happens in the setting of proximity spaces. While a parallel theory for uniform and strong uniform convergence was recently developed, and a notion of proximal convergence is present in the literature, the notion of strong proximal convergence was never considered. In this paper, we propose several possible convergence notions, and we provide complete comparisons among these concepts and the notion of strong uniform convergence in uniform spaces. It is also shown that in particularly meaningful classes of functions these notions are equivalent and can be considered as natural definitions of strong proximal convergence. Finally we consider a function acting between two proximity spaces and we connect its continuity/strong continuity to convergence in the respective hyperspaces of a natural functor associated to the function itself

Graph topologies on closed multifunctions

[EN] In this paper we study function space topologies on closed multifunctions, i.e. closed relations on X x Y using various hypertopologies. The hypertopologies are in essence, graph topologies i.e topologies on functions considered as graphs which are subsets of X x Y . We also study several topologies, including one that is derived from the Attouch-Wets filter on the range. We state embedding theorems which enable us to generalize and prove some recent results in the literature with the use of known results in the hyperspace of the range space and in the function space topologies of ordinary functions.Di Maio, G.; Meccariello, E.; Naimpally, S. (2003). Graph topologies on closed multifunctions. Applied General Topology. 4(2):445-465. doi:10.4995/agt.2003.2044.SWORD4454654