Comparison of Convergences for Multifunctions

Abstract

In this paper, we extend the concept of continuous convergence for singlevalued functions to multifunctions and compare it with topological convergence in points, topological convergence in graphs, quasiuniform convergence and almost quasiuniform convergence. Relationships among these kinds of convergences are established and some of results from [3], [9], [11] and [14] are generalized. IntroductionThe concept of convergence of functions is indispensable in both analysis and topology. The purpose of this paper is to compare several types of convergences for multifunctions which have appeared in recent years.Let X and Y be two topological spaces. A subset A of X is said to be a-paracompact

On continuous convergence of nets of multifunctions

“…This paper expands the classical concept of the continuous convergence of nets of multifunctions introduced by Cao, Reilly and Vamanamurthy in [7]. We introduce some new types of properties of convergence of such nets which guarantee the upper or lower semicontinuity of the limit multifunction.…”

“…In [2] and [7], the Hausdorff quasi-uniformly V-convergence was called almost quasiuniformly convergence. In [6], the Hausdorff V-convergence was called quasiuniformly convergence.…”

“…In [6], the Hausdorff V-convergence was called quasiuniformly convergence. Let us remark also that in [17], the lower (resp.…”

A study of function space topologies for multifunctions

“…Our study here is purely topological, unlike [11], where metric spaces and normed spaces are considered for similar results. Similarly, the continuous convergence introduced in our paper is different from that of [2] and [18]. In [2] and [18], upper and lower topologies, defined on the second space, are used for defining continuous convergence.…”

“…Similarly, the continuous convergence introduced in our paper is different from that of [2] and [18]. In [2] and [18], upper and lower topologies, defined on the second space, are used for defining continuous convergence. However our definition is more straight forward and appears similar to its counterpart of single-valued functions.…”