一致覆盖和度量空间的紧映象

本文利用一致覆盖的概念,讨论了度量空间的序列覆盖紧映象的结构.主要结果有: (1)空间X是局部可分度量空间的序列覆盖紧映象当且仅当X具有由cosmic子空间构成的一致sn网; (2)空间X是局部可分度量空间的序列覆盖,商紧映象当且仅当X是度量空间的序列覆盖,商紧映象且是局部cosmic空间.     

局部可分度量空间闭s映象的注记

该文讨论局部可分度量空间闭s映象的分解定理, 证明了正则的Fréchet空间是局部可分度量空间的闭s映象当且仅当满足如下条件: 具有点可数的cs*网, 第一可数的闭子空间是局部可分的, 且Lindelof的闭子空间是可分的.     

The Kripke schema in metric topology

A form of Kripke's schema turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke's schema serves as a point of reference for classifying theorems of classical mathematics within Bishop‐style constructive reverse mathematics.     

Dense computability structures

We examine computability structures on a metric space and the relationships between maximal, separable and dense computability structures. We prove that in a computable metric space which has the effective covering property and compact closed balls for a given computable sequence which is a metric basis there exists a unique maximal computability structure which contains that sequence. Furthermore, we prove that each maximal computability structure on a convex subspace of Euclidean space is dense. We also examine subspaces of Euclidean space on which each dense maximal computability structure is separable and prove that spheres, boundaries of simplices and conics are such spaces.     

关于序列覆盖s映射的注记

本文分别给出局部可分度量空间的强序列覆盖（1序列覆盖,2序列覆盖）。映象的新刻画,还分别给出拓扑空间是局部可分度量空间的序列覆盖（紧覆盖）s映象的一个充分条件．     

On completion of fuzzy metric spaces

Completions of fuzzy metric spaces (in the sense of George and Veeramani) are discussed. A complete fuzzy metric space Y is said to be a˜fuzzy metric completion of a˜given fuzzy metric space X if X is isometric to a˜dense subspace of Y. We present an example of a˜fuzzy metric space that does not admit any fuzzy metric completion. However, we prove that every standard fuzzy metric space has an (up to isometry) unique fuzzy metric completion. We also show that for each fuzzy metric space there is an (up to uniform isomorphism) unique complete fuzzy metric space that contains a˜dense subspace uniformly isomorphic to it.     

Sigma-compact,nowhere locally compact metric spaces can be densely imbedded in Hilbert space

It is shown that if H is a connected, locally contractible, separable, topologically complete metric space with the property that mappings of separable metric spaces into H are approximable by imbeddings (in particular, if H is Hilbert space), then every sigma-compact, nowhere locally compact metric space can be densely imbedded in H.     

Exchangeable processes need not be mixtures of independent,identically distributed random variables

Summary According to a theorem of de Finetti's, an exchangeable stochastic process with values in a compact metric space can be represented as a mixture of sequences of independent, identically distributed random variables. This paper demonstrates the existence of a separable metric space for which the conclusion fails. In the opposite direction, an example is given of a nonstandard space for which the representation necessarily holds.Modifications of the argument lead to examples of exchangeable stochastic processes and stationary Markov processes which take values in a separable metric space but do not satisfy the conclusions of the Kolmogorov consistency theorem.Research partially supported by National Science Foundation Grant MCS 77-01665     

Fuzzy metric spaces

In this paper, fuzzy metric spaces are redefined, different from the previous ones in the way that fuzzy scalars instead of fuzzy numbers or real numbers are used to define fuzzy metric. It is proved that every ordinary metric space can induce a fuzzy metric space that is complete whenever the original one does. We also prove that the fuzzy topology induced by fuzzy metric spaces defined in this paper is consistent with the given one. The results provide some foundations for the research on fuzzy optimization and pattern recognition.     

区间值模糊集之间一种新距离的完备性

定义区间值模糊集之间的一种新距离d*∞,讨论了这种距离的基本性质,并证明区间值模糊数的距离空间(IΦ*(R),d*∞)是完备的,但它却是不可分的。     

On fuzzy metric spaces

In this paper we introduce the concept of a fuzzy metric space. The distance between two points in a fuzzy metric space is a non-negative, upper semicontinuous, normal and convex fuzzy number. Properties of fuzzy metric spaces are studied and some fixed point theorems are proved.     

EXISTENCE THEOREMS FOR RANDOM MEASURES

Two existence theorems of random measures on a separable complete metric space areproved.It seems that the theory of random measures in locally compact spaces and that inseparable complete metric spaces are essentially different by noting that the critera fortightness of the locally finite measures is much more tedious than that of the Radommeasures.     

星可数k网与序列覆盖s映射

证明了在空间具有星可数ｋ网的条件下,度量空间的１（２）序列覆盖ｓ映象是局部可分度量空间的１（２）序列覆盖、紧覆盖ｓ映象。     

Fuzzy n-cell numbers and the differential of fuzzy n-cell number value mappings

In this paper, we give the definition of a special kind of n-dimension fuzzy numbers, fuzzy n-cell numbers, discuss their operations and representation theorems, define a complete metric on the fuzzy n-cell number space and prove that the metric is equivalent to the supremum metric derived by the Hausdorff metric between the level sets of the n-dimension fuzzy numbers, and obtain an embedding theorem of the fuzzy n-cell number space (isometrically embeds it into a concrete Banach space). We also consider the differential of the fuzzy mappings from an interval into the fuzzy n-cell number space by using the embedding theorem.     

模糊粒度空间的性质研究

在文献[16]基础上,进一步将模糊粒度空间推广到更一般地模糊等价关系上,研究了模糊粒度空间的性质,主要获得了3个结论.首先,引入了有序的等价关系集的概念,给出了下列的四个命题是等价的:(1) 给定一个模糊等价关系;(2) 给定一个等腰归一化伪距离;(3) 给定一个有序的粒度空间;(4) 给定一个有序的等价关系集.第二,通过模糊等价关系诱导的等腰归一化伪距离的投影距离和扩展距离,建立了模糊粒度空间上的距离,即是等腰归一化距离,并且给出了模糊粒度空间上距离度量的动态性质研究.最后,给出了模糊粒度空间与模糊等价关系之间的序关系,即它们的序是一致的.这些研究工作进一步完善了模糊粒度空间的理论,为模糊粒度计算提供了更为直观的数学理论和工具.     

Controlled fuzzy metric spaces and some related fixed point results

In this paper, we introduce a new extension in the subject of fuzzy metric, called controlled fuzzy metric space. This notion is a generalization of fuzzy b‐metric space. Also, we prove a Banach‐type fixed point theorem and a new fixed point theorem for some self‐mappings satisfying fuzzy ψ ‐contraction condition that is more general than existing theorems. Furthermore, we establish some examples about our main results.     

局部可分度量空间的序列覆盖s象

本文给出了局部可分度量空间的1序列覆盖s象,2序列覆盖s象,强序列覆盖s象,序列覆盖s象及子序列覆盖s象的内在刻划．从而使关于对度量空间的各类s象的内在刻划方面的研究更趋于完整．     

Metric spaces in synthetic topology

We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree.     

Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls

We show that the open unit ball of the space of operators from a finite-dimensional Hilbert space into a separable Hilbert space (we call it “operator ball”) has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In Appendix A we present results of Itai Shafrir about hyperbolic metrics on the operator ball.