Regular convergence

Regular Convergence

ON r-REGULAR CONVERGENCE

In his paper On sequences and limiting sets [ l ] , 1 G. T. Whyburn introduced the notion of regular convergence. He showed that in the cases of 0 and 1 regular convergence (see definition below) that the limit of sequences of many simple topological sets is of the same type as the members of the sequence. I t is the purpose of this paper to extend some of these results to higher dimensions. Th...

Regular Convergence of Manifolds with Boundary

In the author's paper [l ] it is shown that if a sequence of orientable »-dimensional generalized closed manifolds (abbreviated »-gem) converge (« —1)-regularly to an »-dimensional set, then the limit set is also an orientable »-gem (see Definitions 1 and 2). In this paper a similar result is obtained for manifolds with boundary. Throughout the paper we assume that our sets are imbedded in a co...

The Convergence Determining Class of Regular Open Sets

The purpose of this paper is to prove that every sequence of closed approximable measures defined on the Borelfield of a normal topological space with values in an abelian topological group is Cauchy convergent for all Borel sets if it is Cauchy convergent for all regular open sets. In particular every sequence of measures on the Borel-field of a perfectly normal topological space which is Cauc...

Convergence of the Proximal Point Method for Metrically Regular Mappings

In this paper we consider the following general version of the proximal point algorithm for solving the inclusion T (x) 3 0, where T is a set-valued mapping acting from a Banach space X to a Banach space Y . First, choose any sequence of functions gn : X → Y with gn(0) = 0 that are Lipschitz continuous in a neighborhood of the origin. Then pick an initial guess x0 and find a sequence xn by appl...