BEST PROXIMITY PAIR THEOREM IN METRIZABLE TOPOLOGICAL VECTOR SPACES

The object of this paper is to prove an existence result on best proximity pair. For this purpose, the class of factorizable multifunctions in approximately weakly compact, convex subset of metrizable topological vector space is used. As consequence, our theorem generalizes the result of Basha and Veeramani. Finally, certain known results have also been obtained as corollaries in this work.

METRIZABILITY OF SPACES AND J.NAGATA'S PROBLEM

In 1988, J.Nagata raised a problem about metrizability, The present article is an attempt to find a solution to the problem.

SOME GENERALIZATIONS OF KY FAN'S BEST APPROXIMATION THEOREM

We pressent new Ky Fan type best approximation theorems for a discontinuous multivalued map on metrizable topological vector spaces and hyperconvex spaces. In addition, fixed point results are derived for the map studied. Our work generalizes severl results in approximation theory.

cs＊-regular Families and Metrization Theorems

In this paper, we proved that a sequential space has a cs＊-regular cs＊-network （or a cs＊-regular weak base） is metrizable, which generalized related results in [6], [12] and [14].

More Properties of Semi-Linear Uniform Spaces

In this paper we shall generalize the definition given in [1] for Lipschitz condition and contractions for functions on a non-metrizable space, besides we shall give more properties of semi-linear uniform spaces.

Two Notes on Topological Groups

In this short paper, we firstly give a short proof of Birkhoff-Kakutani Theorem by Moore metrizable Theorem. Then we prove that G is a topological group if it is a paratopological group which is a dense G_δ-set in a locally feebly compact regular space X.

Equicontinuous Seton Locally Convex Spaces

Let (E,γ) be a locally convex space and E′ its conjugate space. AE′ be an equicontinuous set on (E,γ). In this paper,we show that for each sequence {f i}A and {x j}E, if {x j} converges to 0 in (E,γ), then we can find a f 0∈E′ and extract subsequences {f n i } and {x n j } such that {f n i } converges to f 0 on {x n j } uniformly. If (E,γ) is metrizable,then we can show that the converse is also valid.

Some questions on partial metric spaces

We show that the completion of a partial metric space can fail be unique,which answers a question on completions of partial metric spaces.In addition,to this paper discusses metrizability around partial metric spaces.

Metrization of Topological Spaces with a cs-regular Base

In this paper,we introduce the notions of a cs * regular base,a cs * development and the discretely cs * θ expandability for topological spaces and use these notions to give some metrization theorems. That is a generaliztion of several theorems in [1].

THE THEORY AND APPLICATIONS OF GENERALIZED H-SPACE

This paper brings forward the concept of generalized H-spaces which extends the concepts of H-spaces and almost probabilistic metric spaces. In this paper, the uniformity and properties far generalized H-space are considered. The conditions of metrization and the form of metric functions for generalized H-spaces, H-spaces and Menger PM-spaces are given and the characteristics of completeness and compactness for generalized H-spaces are presented. The results of this paper generalize and unify some recent results of [1-2, 8, 10].

Semantic model and optimization of creative processes at mathematical knowledge formation

The aim of this work is mathematical education through the knowledge system and mathematical modeling. A net model of formation of mathematical knowledge as a deductive theory is suggested here. Within this model the formation of deductive theory is represented as the development of a certain informational space, the elements of which are structured in the form of the orientated semantic net. This net is properly metrized and characterized by a certain system of coverings. It allows injecting net optimization parameters, regulating qualitative aspects of knowledge system under consideration. To regulate the creative processes of the formation and realization of mathematical know- edge, stochastic model of formation deductive theory is suggested here in the form of branching Markovian process, which is realized in the corresponding informational space as a semantic net. According to this stochastic model we can get correct foundation of criterion of optimization creative processes that leads to “great main points” strategy (GMP-strategy) in the process of realization of the effective control in the research work in the sphere of mathematics and its applications.

Researches on point-discrete families

Based on the modern development of Metrization theorem for context,the main results obtained in recent ten years on point-discrete families are summarized.This paper mainly introduces the theory of the spaces with opoint-discrete bases,the spaces with certain o-point-discrete networks,and the relationship between the above spaces and the spaces with certain o-compactfinite networks.

On CWC-Mappings and Metrization Theorem

In this paper,applications of CWC-mappings are discussed in metrization theorem, and some new metrization theorems of topological spaces are obtained by Nagata's condition and weakγ-spaces,which generalizes several metrization theorems obtained by R.E.Hodel and J.Nagata.

Topological algebras of infinite matrix operators between sequence spaces λ and μ

SINCE Michael (1952) published the first book about topological algebras, it has become abranch of functional analysis. Applications of topological algebras have been found in complexanalysis of several variables, differential geometry, unbounded operator and others, and alge-braic topology, k-theory and others have been applied to topological algebras (see, for exam-ple, ref.[2]). It is well known that the set of all continuous linear operators on Banach spaceconstitutes a Banach algebra, so it is obviously meaningful to study the topological algebra con-stituted by the set of all continuous linear operators on concrete topological vector space. It

A trichotomy for a class of equivalence relations

Let Xn, n ∈ N be a sequence of non-empty sets, ψn : Xn2 → IR+. We consider the relation E = E((Xn, ψn)n∈N) on ∏n∈N Xn by (x, y) ∈ E((Xn, ψn)n∈N) <=>Σn∈Nψn(x(n), y(n)) < +∞. If E is an equiv- alence relation and all ψn, n ∈ N, are Borel, we show a trichotomy that either IRN/e1≤B E, E1≤B E, or E≤B E0. We also prove that, for a rather general case, E((Xn, ψn)n∈N) is an equivalence relation iff it is an ep-like equivalence relation.