Weak regularity and consecutive topologizations and regularizations of pretopologies

L. Foged proved that a weakly regular topology on a countable set is regular. In terms of convergence theory, this means that the topological reflection Tξ of a regular pretopology ξ on a countable set is regular. It is proved that this still holds if ξ is a regular σ-compact pretopology. On the other hand, it is proved that for each n<ω there is a (regular) pretopology ρ (on a set of cardinality c) such that k(RT)ρ>n(RT)ρ for each k<n and n(RT)ρ is a Hausdorff compact topology, where R is the reflector to regular pretopologies. It is also shown that there exists a regular pretopology of Hausdorff RT-order ?ω₀. Moreover, all these pretopologies have the property that all the points except one are topological and regular.     

Projective versions of selection principles

All spaces are assumed to be Tychonoff. A space X is called projectively P (where P is a topological property) if every continuous second countable image of X is P. Characterizations of projectively Menger spaces X in terms of continuous mappings , of Menger base property with respect to separable pseudometrics and a selection principle restricted to countable covers by cozero sets are given. If all finite powers of X are projectively Menger, then all countable subspaces of Cp(X) have countable fan tightness. The class of projectively Menger spaces contains all Menger spaces as well as all σ-pseudocompact spaces, and all spaces of cardinality less than d. Projective versions of Hurewicz, Rothberger and other selection principles satisfy properties similar to the properties of projectively Menger spaces, as well as some specific properties. Thus, X is projectively Hurewicz iff Cp(X) has the Monotonic Sequence Selection Property in the sense of Scheepers; βX is Rothberger iff X is pseudocompact and projectively Rothberger. Embeddability of the countable fan space Vω into Cp(X) or Cp(X,2) is characterized in terms of projective properties of X.     

Densely Continuous Forms in Vietoris Hyperspaces

For countably paracompact normal spaces X and locally compact separable metric spaces Y, a characterization is given for the closure of the set of densely continuous forms from X to Y in the hyperspace of nonempty closed subsets of X × Y under the Vietoris topology. This shows that for such X having no isolated points, every closed subset of X × R that is dense over X can be Vietoris approximated by a semicontinuous function on X.     

Semiconductor Equations for variable Mobilities Based on Boltzmann Statistics or Fermi-Dirac Statistics

The first aim of this paper is to characterize those limit spaces (X, τ) which can be valuated, i.e. for which a set E of valuations exists such that for each x ∈ X, τ(x) equals the set of filters on X which converge to x relative to E. It is shown further that a separated pretopological space is a URYSOHN -space iff it can be valuated by a set of subadditive valuations. In the second part of the paper a completion is constructed for the separated subadditively valuated limit spaces which can be considered as a generalization of the usual completion of a uniform space.     

Remotality of Closed Bounded Convex Sets in Reflexive Spaces

Let X be a Banach space and E be a closed bounded subset of X. For x ? X, we define D(x, E) = sup{‖ x ? e‖:e ? E}. The set E is said to be remotal (in X) if, for every x ? X, there exists e ? E such that D(x, E) = ‖x ? e‖. The object of this paper is to characterize those reflexive Banach spaces in which every closed bounded convex set is remotal. Such a result enabled us to produce a convex closed and bounded set in a uniformly convex Banach space that is not remotal. Further, we characterize Banach spaces in which every bounded closed set is remotal.     

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)     

一类分数次次线性算子及其交换子在齐型空间上的弱Morrey-Herz空间上的有界性

本文研究了一类次线性算子及其交换子在齐型空间上的弱有界性的问题.利用齐型空间的基本性质以及给出的一类次线性算子及其分别与BMO函数,Lipschitz函数生成的交换子在L~p(X)上的弱有界性,证明了其在齐型空间上Morrey-Herz空间中的弱有界性.推广了该类算子在Morrey-Herz空间中的强有界性这一结果.     

Regularity properties of transition probabilities in infinite dimensions

In a Hilbert space H we consider a process X solution of a semilinear stochastic differential equation, driven by a Wiener process. We prove that, under appropriate conditions, the transition probabilities of X are absolutely continuous with respect to a properly chosen gaussian measure μ in H, and the corresponding densities belong to some Wiener-Sobolev spaces over (H,μ). In the linear caseX is a nonsymmetric Ornstein-Uhlenbeck process, with possibly degenerate diffusion coefficient. The general case is treated by the Girsanov. Theorem and the Malliavin calculus. Examples and applications to stochastic partial differential equations are given     

Vector lattice covers of ideals and bands in Pre-Riesz spaces

Abstract

Pre-Riesz spaces are ordered vector spaces which can be order densely embedded into vector lattices, their so-called vector lattice covers. Given a vector lattice cover Y for a pre-Riesz space X, we address the question how to find vector lattice covers for subspaces of X, such as ideals and bands. We provide conditions such that for a directed ideal I in X its smallest extension ideal in Y is a vector lattice cover. We show a criterion for bands in X and their extension bands in Y as well. Moreover, we state properties of ideals and bands in X which are generated by sets, and of their extensions in Y.     

Betti Numbers of parabolic U(2,1)-Higgs bundles moduli spaces

Let X be a compact Riemann surface together with a finite set of marked points. We use Morse theoretic techniques to compute the Betti numbers of the parabolic U(2,1)-Higgs bundles moduli spaces over X. We give examples for one marked point showing that the Poincaré polynomials depend on the system of weights of the parabolic bundle.      

A Banach–Stone Theorem for Uniformly Continuous Functions

 In this note we prove that the uniformity of a complete metric space X is characterized by the vector lattice structure of the set U(X) of all uniformly continuous real functions on X.     

关于“预拓扑与网族的预收敛类”的注

本文研究了预拓扑空间中的收敛问题.利用完备格同构的方法,获得了预拓扑与预收敛类可以相互确定的结果,推广了拓扑与收敛类可以相互确定的结果,同时推广了文献[1]的结果.     

Research announcement on gaierkirfs method for a class of integra equations of the first kind

Let X and Y be locally convex spaces with K a closed convex cone in X Necessary and sufficient conditions are given for the image AK to be closed in Ywhen A:X→Y is a continuous linear map. This result is used to generalize a theorem of Abrams to infinite dimensional spaces and also to give sufficient conditions for the Hurwicz version of the Farkas lemma for locally convex spaces to hold.     

Cyrtologies of Convergences,I

We study the complete lattice of convergences, that is, of relations between a set and the set of its filters. Various properties of convergences (like isotonicity, antitonicity, being a pseudotopology, a pretopology, an adherence) determine subsets of the lattice of convergences which possess certain particular features called “cyrtological”. The morphisms associated to these sets, said “cyrtomorphisms”, are basic in the study of convergences. A method of generation of GALOIS connections relative to cyrtomorphisms is proposed. Semigroups generated by some cyrtomorphisms are analyzed.     

Reiteration formulae for interpolation methods associated to polygons

We study spaces generated by applying the interpolation methods defined by a polygon Π to an N-tuple of real interpolation spaces with respect to a fixed Banach couple {X,Y}. We show that if the interior point (α,β) of the polygon does not lie in any diagonal of Π then the interpolation spaces coincide with sums and intersections of real interpolation spaces generated by {X,Y}. Applications are given to N-tuples formed by Lorentz function spaces and Besov spaces. Moreover, we show that results fail in general if (α,β) is in a diagonal.     

Spectrum topology of a residuated lattice

In this paper, given a non-commutative residuated lattice L, a topological space is constructed using certain fuzzy subsets of L. Indeed, we show that the set of all prime fuzzy filters of a non-commutative residuated lattice L forms a topological space. Particularly, we show that this space is compact and a T ₀-space and its certain subspaces are Hausdorff spaces. Finally, we show that the set of all prime filters of L is also a Hausdorff space.     

Isomorphisms of lattices of subalgebras of semirings of continuous nonnegative functions

In this work, lattice isomorphisms of semirings C ⁺(X) of continuous nonnegative functions over an arbitrary topological space X are characterized. It is proved that any isomorphism of lattices of all subalgebras with a unit of semirings C ⁺(X) and C ⁺(Y) is induced by a unique isomorphism of semirings. The same result is also correct for lattices of all subalgebras excepting the case of two-point Tychonovization of spaces.     

Bilipschitz Embeddings of Metric Spaces into Space Forms

The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.     

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.