On set functions with values in a topological vector space

Summary Spaces of set functions defined on a σ-ring and taking values in a topological vector space are supplied with a certain weak topology. Their continuous duals are characterized; certain subsets (the set of simple measures) are singled out and studied and compactness conditions are given. Finally the properties of metrizability, normality, barreledness and of being semi-Montel are studied in connection with these spaces. Entrata in Redazione il 12 maggio 1971.     

Group duality with the topology of precompact convergence

The Pontryagin-van Kampen (P-vK) duality, defined for topological Abelian groups, is given in terms of the compact-open topology. Polar reflexive spaces, introduced by Köthe, are those locally convex spaces satisfying duality when the dual space is equipped with the precompact-open topology. It is known that the additive groups of polar reflexive spaces satisfy P-vK duality. In this note we consider the duality of topological Abelian groups when the topology of the dual is the precompact-open topology. We characterize the precompact reflexive groups, i.e., topological groups satisfying the group duality defined in terms of the precompact-open topology. As a consequence, we obtain a new characterization of polar reflexive spaces. We also present an example of a space which satisfies P-vK duality and is not polar reflexive. Some of our results respond to questions appearing in the literature.     

Permanence properties of C(X,E)

Let C(X,E) be the vector space of all continuous functions on a completely regular Hausdorff space X with values in a locally convex space E, equipped with the compact-open topology. In this note it will be shown that for many classes of locally convex spaces K C(X,E) lies in K if and only if C(X)=C(X,¦K) (¦K=¦R or C) and E belong to K. This is valid for the classes K. of all metrizable, normed, (DF)-, -locally topological, separable, quasi-complete, complete, nuclear, Schwartz, semi-Montel, Montel, semi-reflexive, reflexive and quasi-normable locally convex spaces, respectively. But in general C(X,E) is not quasi-barrelled, barrelled and bornological, respectively, if C(X) and E belong to the same class. We shall give sufficient conditions for C(X,E) to be quasi-barrelled and barrelled, respectively.Herrn Professor Gottfried Köthe zum 75. Geburtstag gewidmet     

A combinatorial characterisation of embedded polar spaces

Some classical polar spaces admit polar spaces of the same rank as embedded polar spaces (often arisen as the intersection of the polar space with a non-tangent hyperplane). In this article we look at sets of generators that behave combinatorially as the set of generators of such an embedded polar space, and we prove that they are the set of generators of an embedded polar space.     

Expanding topological space,study and applications

In this paper, we introduce the notion of expanding topological space. We define the topological expansion of a topological space via local multi-homeomorphism over coproduct topology, and we prove that the coproduct family associated to any fractal family of topological spaces is expanding. In particular, we prove that the more a topological space expands, the finer the topology of its indexed states is. Using multi-homeomorphisms over associated coproduct topological spaces, we define a locally expandable topological space and we prove that a locally expandable topological space has a topological expansion. Specifically, we prove that the fractal manifold is locally expandable and has a topological expansion.     

Invariants of Directed Spaces

Directed spaces are the objects of study within directed algebraic topology. They are characterised by spaces of directed paths associated to a source and a target, both elements of an underlying topological space. The algebraic topology of these path spaces and their connections are studied from a categorical perspective. In particular, we study the preorder category associated to a directed space and various “quotient” categories arising from algebraic topological functors. Furthermore, we propose and study a new notion of directed homotopy equivalence between directed spaces.      

Compact Sets of Holomorphic Mappings

Compactness of locally bounded sets of holomorphic functions with infinite dimensional domains is connected, using Heinrich's density condition, to the Schwartz and semi-Montel properties on the domain. The metrizability of bounded subsets for various spaces of holomorphic functions is investigated.     

Stein domains in Banach algebraic geometry

In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a derived perspective on analytic geometry.     

双曲复空间的拓扑结构与应用

双曲复空间与Minkowski空间相对应,具有时空方向异性的特点。以双曲复空间为原空间,可以抽象出一类双曲拟、虚度量空间和多拓扑结构。     

s-乘数收敛及Orlicz-Pettis型定理

本文给出了在局部凸空间中与弱拓扑具有相同的ｓ－乘数收敛点列的最强的可允许极拓扑Ｆ（μ_ｓ）的刻划．并给出Ｆ（μｓ）＝β（Ｘ,Ｘ＇）的充分条件和必要条件,由此证明了ｃ０（或ｌｐ,０＜ｐ＜∞）－乘数收敛性是对可允许极拓扑全体而言的不变性,     

Some generalizations of Backʼs Theorem

The problem of the existence of jointly continuous utility functions is studied. A continuous representation theorem of Back [1] gives the existence of a continuous map from the space of total preorders topologized by closed convergence (Fell topology) to the space of utility functions with different choice sets (partial maps) endowed with a generalization of the compact-open topology. The commodity space is locally compact and second countable. Our results generalize Back?s Theorem to non-metrizable commodity spaces with a family of not necessarily total preorders. Precisely, we consider regular commodity spaces having a weaker locally compact second countable topology.     

Comparison of Some Classical Topologies on Hyperspaces within the Framework of Point Spaces

S. Dolecki, G. Greco and A. Lechicki call a space X consonant if the co-compact topology and the upper Kuratowski topology on the set of closed subsets of X coincide. We call a space X hyperconsonant if Fell's topology and the (Kuratowski) convergence topology coincide. Recently, we proved that a first countable, locally paracompact, T ₃-space is hyperconsonant if and only if the space possesses at most one point without a compact neighbourhood, extending the same result of D. Fremlin obtained for metrizable spaces. In this paper, we pursue the study of hyperconsonance within the framework of point spaces (countable T ₁-spaces with exactly one accumulation point) and we compare consonance and hyperconsonance in such spaces. In particular, we answer a question of T. Nogura and D. Shakhmatov: does there exist a nonconsonant point space? We provide a Fréchet, -point space which is not consonant. Moreover, this example proves that the consonance is not preserved by continuous closed compact-covering maps of separable complete metrizable spaces onto Hausdorff spaces.     

Strict and Mixed Topologies on Function Spaces

In this paper we prove that the strict topology on spaces of continuous and holomorphic functions on a BANACH space can be considered a mixed topology. Using this fact, we obtain new results about the strict topology, as an application of the general properties of mixed topologies.     

Separated totally convex spaces

In this paper we define for every totally convex space a suitable topology, the radial topology. We prove that a morphism in the category TC^sep of separated totally convex spaces is an epimorphism if and only if its image is dense in the radial topology, and that TC^sep is the full subcategory of TC generated by its Hausdorff objects. These results remain valid for finitely totally convex spaces when the radial topology is replaced by the distance-radial topology.Dedicated to Karl Stein     

Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows—provided the sequence is tight—from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Λ-coalescents. We show that the Λ-coalescent defines an infinite (random) metric measure space if and only if the so-called “dust-free”-property holds.     

Polar Foliations of Symmetric Spaces

We prove that a polar foliation of codimension at least three in an irreducible compact symmetric space is hyperpolar, unless the symmetric space has rank one. For reducible symmetric spaces of compact type, we derive decomposition results for polar foliations.     

Quasi-metrics and monotone normality

The purpose of this paper is to study which quasi-metrizable spaces are monotonically normal. In particular, we provide a sufficient condition for a quasi-metrizable space to be monotonically normal. This enables us to prove the monotone normality of a certain amount of interesting examples of quasi-metric spaces; for instance, we show that the continuous poset of formal balls of a metric space, endowed with the Scott topology, is a monotonically normal quasi-metrizable space.     

A Note on Sequential Properties of Noncompactness in Banach Spaces

We define two properties of sequences in Banach spaces that may be related to measures of noncompactness of subsets of these spaces. The first one concerns properties of sequences related to the strong topology, and the second one is related to the weak topology. Given a Banach space X, we introduce a new Banach space such that we can find a subset E in it that may be identified with the balls in the first one. We use compactness in this new space to characterize our sequential properties. In particular, we prove a general form of the Eberlein-Smulian theorem. This revised version was published online in June 2006 with corrections to the Cover Date.     

集值映射空间在紧开拓扑下的N_0性质

本文讨论了点紧致的连续集值映射空间在赋予紧开拓扑下的某些拓扑性质,证明了:若X,Y为N_0空间,则X到Y上的点紧致的连续集值映射族依紧开拓扑是N_0空间,从而将Michael的结论推广到更大的映射空间类上.     

Topologies on function spaces

In the present paper we introduce notions of A-splitting and A-jointly continuous topology on the set C(Y,Z) of all continuous maps of a topological space Y into a topological space Z, where A is any family of spaces. These notions satisfy the basic properties of splitting and jointly continuous topologies on C(Y,Z). In particular, for every A, the greatest A-splitting topology on C(Y,Z) (denoted by τ(A) always exists. We indicate some families A of spaces for which the topology τ(A) coincides with the greatest splitting topology on C(X,Y). We give a notion of equivalent families of spaces and try to find a “simple” family which is equivalent to a given family. In particular, we prove that every family is equivalent to a family consisting of one space, and the family of all spaces is equivalent to a family of all T₁-spaces containing at most one nonisolated point. We compare the topologies τ({X}) for distinct compact metrizable spaces X and give some examples. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 82–97. Translated by A. A. Ivanov.