A topology for galois types in abstract elementary classes

We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model‐theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

EQUILIBIA OF NONCOMPACT GENERALIZED GAMES WITH CORRESPONDENCES WITHOUT OPEN LOWER SECTIONS

1IntroductionOneofthemainresultsofequilibriullltheoryforgelleralizedgalllesistileexistellceofmaximalelementsconcerninganirreflexivepreferencecorrespondellcewitllcollvexvalues,openlowersections,definedonacompactconvexsubsetKofRnalldwithvaluesillZap,seen.Chapter7].Duetotheimportallceofapplications,tileresultwasgelleralizedby1flailyauthors,forexample,see[2,5--7,10,13,15-181.Tilelllostoftheseexistellcetheorellisforlllaxilllalelementsandequilibriulllpointsdealwithpreferencecorrespondenceswhichhav…     

Double convergence and products of Fréchet spaces

The paper is devoted to convergence of double sequences and its application to products. In a convergence space we recognize three types of double convergences and points, respectively. We give examples and describe their structure and properties. We investigate the relationship between the topological and convergence closure product of two Fréchet spaces. In particular, we give a necessary and sufficient condition for the topological product of two compact Hausdorff Fréchet spaces to be a Fréchet space.     

Interior operators and topological separation

A notion of separation with respect to an interior operator in topology is introduced and some basic properties are presented. In particular, it is shown that this notion of separation with respect to an interior operator gives rise to a Galois connection between the collection of all subclasses of the class of topological spaces and the collection of all interior operators in topology. Characterizations of the fixed points of this Galois connection are given and examples are provided.     

Continuous group actions on profinite spaces

For a profinite group, we construct a model structure on profinite spaces and profinite spectra with a continuous action. This yields descent spectral sequences for the homotopy groups of homotopy fixed point spaces and for stable homotopy groups of homotopy orbit spaces. Our main example is the Galois action on profinite étale topological types of varieties over a field. One motivation is to understand Grothendieck’s section conjecture in terms of homotopy fixed points.     

Generalized Topological Essentiality and Coincidence Points of Multivalued Maps

A concept of generalized topological essentiality for a large class of multivalued maps in topological vector Klee admissible spaces is presented. Some direct applications to differential equations are discussed. Using the inverse systems approach the coincidence point sets of limit maps are examined. The main motivation as well as main aim of this note is a study of fixed points of multivalued maps in Fréchet spaces. The approach presented in the paper allows to check not only the nonemptiness of the fixed point set but also its topological structure.      

A GALOIS CORRESPONDENCE BETWEEN CLOSURE SPACES AND RELATIONAL SYSTEMS

Abstract

For each ordinal α < 1 we define a Galois correspondence between the category of closure spaces (i.e. sets endowed with an extensive and monotone closure operation taking value θ at θ) and the category of α-ary mono-relational systems. These correspondences are studied.     

Integral representation of belief measures on compact spaces

An integral representation theorem for outer continuous and inner regular belief measures on compact topological spaces is elaborated under the condition that compact sets are countable intersections of open sets (e.g. metric compact spaces). Extreme points of this set of belief measures are identified with unanimity games with compact support. Then, the Choquet integral of a real valued continuous function can be expressed as a minimum of means over the sigma-core and also as a mean of minima over the compact subsets. Similarly, for bounded measurable functions, the Choquet integral is expressed as min of means over the core, we prove in addition that it is a mean of infima over the compact subsets. Then, we obtain Choquet–Revuz' measure representation theorem and introduce the Möbius transform of a belief measure. An extension to locally compact and sigma-compact topological spaces is provided.     

Arzelà's Theorem and strong uniform convergence on bornologies

In 1883 Arzelà (1983/1984) [2] gave a necessary and sufficient condition via quasi-uniform convergence for the pointwise limit of a sequence of real-valued continuous functions on a compact interval to be continuous. Arzelà's work paved the way for several outstanding papers. A milestone was the P.S. Alexandroff convergence introduced in 1948 to tackle the question for a sequence of continuous functions from a topological space (not necessarily compact) to a metric space. In 2009, in the realm of metric spaces, Beer and Levi (2009) [10] found another necessary and sufficient condition through the novel notion of strong uniform convergence on finite sets. We offer a direct proof of the equivalence of Arzelà, Alexandroff and Beer-Levi conditions. The proof reveals the internal gear of these important convergences and sheds more light on the problem. We also study the main properties of the topology of strong uniform convergence of functions on bornologies, initiated in Beer and Levi (2009) [10].     

On convergence theory in fuzzy topological spaces and its applications

In this paper we introduce and study new concepts of convergence and adherent points for fuzzy filters and fuzzy nets in the light of the Q-relation and the Q-neighborhood of fuzzy points due to Pu and Liu [28]. As applications of these concepts we give several new characterizations of the closure of fuzzy sets, fuzzy Hausdorff spaces, fuzzy continuous mappings and strong Q-compactness. We show that there is a relation between the convergence of fuzzy filters and the convergence of fuzzy nets similar to the one which exists between the convergence of filters and the convergence of nets in topological spaces.     

Approximation of common fixed points of weakly compatible pairs using the Jungck iteration

We show that the Jungck iteration scheme can be used to approximate the common fixed points of some weakly compatible pairs of generalized quasicontractive operators defined on metric spaces. The existence of coincidence points are also discussed for those pair of maps. The results are generalizations of well known results of the convergence of Picard iterations for single self maps of Banach spaces. In particular, the results improve, generalize and extend the recent results of Berinde [V. Berinde, A common fixed point theorem for compatible quasi contractive self mappings in metric spaces, Applied Mathematics and Computation 213 (2009) 348-354] and answers the open question posed in the paper.     

Axioms for Sequential Convergence

It is of general knowledge that those (ultra)filter convergence relations coming from a topology can be characterized by two natural axioms. However, the situation changes considerable when moving to sequential spaces. In case of unique limit points Kisyński (Colloq Math 7:205–211, 1959/1960) obtained a result for sequential convergence similar to the one for ultrafilters, but the general case seems more difficult to deal with. Finally, the problem was solved by Koutnik (Closure and topological sequential convergence. In: Convergence Structures 1984 (Bechyně, 1984). Math. Res., vol. 24, pp. 199–204. Akademie-Verlag, Berlin, 1985). In this paper we present an alternative approach to this problem. Our goal is to find a characterization more closely related to the case of ultrafilter convergence. We extend then the result to characterize sequential convergence relations corresponding to Fréchet topologies, as well to those corresponding to pretopological spaces.      

The Ultrafilter Closure in ZF

It is well known that, in a topological space, the open sets can be characterized using ?lter convergence. In ZF (Zermelo‐Fraenkel set theory without the Axiom of Choice), we cannot replace filters by ultrafilters. It is proven that the ultra?lter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ultra?lter Theorem is equivalent to the fact that u_X = k_X for every topological space X, where k is the usual Kuratowski closure operator and u is the Ultra?lter Closure with u_X (A):= {x ∈ X: (? U ultrafilter in X)[U converges to x and A ∈ U ]}. However, it is possible to built a topological space X for which u_X ≠ k_X, but the open sets are characterized by the ultra?lter convergence. To do so, it is proved that if every set has a free ultra?lter, then the Axiom of Countable Choice holds for families of non‐empty finite sets. It is also investigated under which set theoretic conditions the equality u = k is true in some subclasses of topological spaces, such as metric spaces, second countable T0‐spaces or {?} (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some properties of bornological convergences

We study some basic properties of the so-called bornological convergences in the realm of quasi-uniform spaces. In particular, we revisit the results about when these convergences are topological by means of the use of pretopologies. This yields a presentation of the bornological convergences as a certain kind of hit-and-miss pretopologies. Furthermore, we characterize the precompactness and total boundedness of the natural quasi-uniformities associated to these convergences. We also obtain an extension of the classical result of Künzi and Ryser about the compactness of the topology generated by the Hausdorff quasi-uniformity to this framework.     

Sobriety in Terms of Nets

Sobriety is a subtle notion of completeness for topological spaces: A space is sober if it may be reconstructed from the lattice of its open subsets. The usual criterion to check sobriety involves either irreducible closed subsets or completely prime filters of open sets. This paper provides an alternative possibility, thus trying to make sobriety easier to understand. We define the notion of observative net, which, together with an appropriate convergence notion, characterizes sobriety. As the filter approach does not involve just usual (topological) convergence, this is not an instance of the classical net-filter translation in general topology.     

Infinite distributive laws versus local connectedness and compactness properties

Various local connectedness and compactness properties of topological spaces are characterized by higher degrees of distributivity for their lattices of open (or closed) sets, and conversely. For example, those topological spaces for which not only the lattice of open sets but also that of closed sets is a frame, are described by the existence of web neighborhood bases, where webs are certain specific path-connected sets. Such spaces are called web spaces. The even better linked wide web spaces are characterized by F-distributivity of their topologies, and the worldwide web spaces (or C-spaces) by complete distributivity of their topologies. Similarly, strongly locally connected spaces and locally hypercompact spaces are characterized by suitable infinite distributive laws. The web space concepts are also viewed as natural extensions of spaces that are semilattices with respect to the specialization order and have continuous (unary, binary or infinitary) semilattice operations.     

On the convergence of constrained minima

We characterize the convergence of values and Lagrange multipliers for minimum problems of perturbed quadratic functionals in Banach spaces subject to a fixed linear operator constraint with finite-dimensional range. We obtain sufficient conditions for the convergence of the solutions. Examples show the different behaviour of the continuous dependence problem for the analogous unconstrained minimizations, where the gamma-type variational convergences are relevant. Such convergence conditions are extended here to the con strained problems.     

Hypertopologies on function spaces

The aim of this paper is to continue Naimpally’s seminal papers [16], [17], [18], i.e. we investigate topological properties of spaces which force the coincidence of convergences of functions associated with different hyperspace topologies. For example a metric spaceX is locally compact iff the topological convergence and the convergence induced by the Fell topology coincide onC(X,IR). Moreover, the proximal topology on the space of functions, not necessarily continuous, is studied in great detail.     

Kodaira fibrations and beyond: methods for moduli theory

Kodaira fibred surfaces are remarkable examples of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. Our tour through algebraic surfaces and their moduli (with results valid also for higher dimensional varieties) deals with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variation of Hodge Structures, especially a recent answer given (in joint work with Dettweiler) to a long standing question posed by Fujita. In the landscape of our tour, Galois coverings, deformations and rigid manifolds (there are by the way rigid Kodaira fibrations), projective classifying spaces, the action of the absolute Galois group on moduli spaces, stand also in the forefront. These questions lead to interesting algebraic surfaces, for instance remarkable surfaces constructed from VHS, surfaces isogenous to a product with automorphisms acting trivially on cohomology, hypersurfaces in Bagnera-de Franchis varieties, Inoue-type surfaces.