Web‐compact spaces,Fréchet‐Urysohn groups and a Suslin closed graph theorem

A topological space X is strongly web‐compact if X admits a family {A_α: α ∈ ?^?} of relatively countably compact sets covering X and such that A_α ? A_β for α ≤ β. The main result of this paper states the following: Theorem A Let X and Y be topological groups and f a homomorphism between X and Y with closed graph. If X is Fréchet‐Urysohn and Baire and Y is strongly web‐compact, then f is continuous. This extends a result of Valdivia. We provide an example showing that the property of being strongly web‐compact is not productive. This applies to show that there are quasi‐Suslin spaces X whose product X × X is not quasi‐Suslin (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

When is a Volterra space Baire?

In this paper, we study the problem when a Volterra space is Baire. It is shown that every stratifiable Volterra space is Baire. This answers affirmatively a question of Gruenhage and Lutzer in [G. Gruenhage, D. Lutzer, Baire and Volterra spaces, Proc. Amer. Math. Soc. 128 (2000) 3115-3124]. Further, it is established that a locally convex topological vector space is Volterra if and only if it is Baire; and the weak topology of a topological vector space fails to be Baire if the dual of the space contains an infinite linearly independent pointwise bounded subset.     

On the compact open and finest splitting topologies

In [M.H. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105-145] it is shown that in the set C(Nω,N) of all continuous maps of Nω into N, where N is an infinitely countable discrete topological space, the compact-open topology is not the finest splitting topology. Since Nω is consonant (see [S. Dolecki, G.H. Greco, A. Lechicki, When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide? Trans. Amer. Math. Soc. 347 (1995) 2869-2884]) the Isbell topology on C(Nω,N) also is not the finest splitting topology. This result is generalized in the present paper proving that it is true also for spaces having the so-called Specific Extension Property. The following spaces have the Specific Extension Property: (a) infinitely countable free unions of non-empty spaces, (b) non-compact Lindelöf zero-dimensional spaces, and (c) metric locally convex linear spaces. In particular, we prove that on the set of all real-valued functions on the (separable infinite dimensional) Hilbert space the compact-open topology does not coincide with the finest splitting topology.     

Weakly K-analytic spaces and the three-space property for analyticity

Let (E,E^′) be a dual pair of vector spaces. The paper studies general conditions which allow to lift analyticity (or K-analyticity) from the weak topology σ(E,E^′) to stronger ones in the frame of (E,E^′). First we show that the Mackey dual of a space Cp(X) is analytic iff the space X is countable. This yields that for uncountable analytic spaces X the Mackey dual of Cp(X) is weakly analytic but not analytic. We show that the Mackey dual E of an (LF)-space of a sequence of reflexive separable Fréchet spaces with the Heinrich density condition is analytic, i.e. E is a continuous image of the Polish space NN. This extends a result of Valdivia. We show also that weakly quasi-Suslin locally convex Baire spaces are metrizable and complete (this extends a result of De Wilde and Sunyach). We provide however a large class of weakly analytic but not analytic metrizable separable Baire topological vector spaces (not locally convex!). This will be used to prove that analyticity is not a three-space property but we show that a metrizable topological vector space E is analytic if E contains a complete locally convex analytic subspace F such that the quotient E/F is analytic. Several questions, remarks and examples are included.     

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.     

[Russian Text Ignored.]

This paper will present some results on quasivariational inequality {C, E, P, Φ} in topological linear locally convex Hausdorff spaces. We shall be concerning with quasivariational inequalities defined on subsets which are convexe closed, or only closed. The compactness of the subset C is replaced by the condensing property of the mapping E. Further, we also obtain some results for quasivariational inequality {C, E, P, Φ}, where the multivalued mapping E maps C into 2^X and satisfies a general inward boundary condition.     

Analytic and Luzin spaces (non-separable case)

We introduce the concept of κ-analytic and κ-Luzin spaces as images of complete metric spaces by (disjoint) upper semi-continuous compact-valued correspondences which “preserve discreteness” in some sence (Definition in Section 3.1). The case κ = ω coincides with (Lindelöf) analytic spaces studied by Choquet, the first author and others. The main results are characterizations of uniform analytic spaces in terms of other parametrizations, complete sequences of covers, and Suslin subsets of some product of a compact and a complete metric space (Theorems in Section 3.2 and in Section 4), and characterizations of topological analytic spaces as Suslin subsets of paracompact ?ech-complete spaces (Theorem in Section 5).     

Local connectedness in merotopic spaces

Abstract

In this paper uniformly locally uniformly connected merotopic spaces are studied. It turns out that their structural behaviour is essentially similar to that one of locally connected topological spaces. The introduced concept is also investigated for spaces of functions between filter-merotopic spaces (e.g. topological spaces, proximity spaces, convergence spaces) and the relationship to other concepts of local connectedness is clarified. In particular, the category of uniformly locally uniformly connected filter-merotopic spaces is Cartesian closed.     

Separating sets by lower and upper semicontinuous functions with a closed graph

In this paper we characterize the pairs (A^?, A⁺) of disjoint subsets of perfectly normal topological space which can be separated by a lower and an upper semicontinuous function with a closed graph.     

LOCAL COMPACTNESS IN SEMI-UNIFORM CONVERGENCE SPACES

Abstract

Local compactness is studied in the highly convenient setting of semi-uniform convergence spaces which form a common generalization of (symmetric) limit spaces (and thus of symmetric topological spaces) as well as of uniform limit spaces (and thus of uniform spaces). It turns out that it leads to a cartesian closed topological category and, in contrast to the situation for topological spaces, the local compact spaces are exactly the compactly generated spaces. Furthermore, a one-point Hausdorff compactification for noncompact locally compact Hausdorff convergence spaces is considered.¹     

Bounded linear regularity of convex sets in Banach spaces and its applications

This paper deals with bounded linear regularity, linear regularity and the strong conical hull intersection property (CHIP) of a collection of finitely many closed convex intersecting sets in Banach spaces. It is shown that, as in finite dimensional space setting (see [6]), the standard constraint qualification implies bounded linear regularity, which in turn yields the strong conical hull intersection property, and that the collection of closed convex sets {C ₁, . . . ,C _n } is bounded linearly regular if and only if the tangent cones of {C ₁, . . . ,C _n } has the CHIP and the normal cones of {C ₁, . . . ,C _n } has the property (G)(uniformly on a neighborhood in the intersection C). As applications, we study the global error bounds for systems of linear and convex inequalities. The work of this author was partially supported by the National Natural Sciences Grant (No. 10471032) and the Excellent Young Teachers Program of MOE, P.R.C The authors thank professor K.F.Ng for his helpful discussion and the referee for their helpful suggestions on improving the first version of this paper     

A note on realizing homomorphisms of category algebras

The complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a complete matrix space X to the category algebra $\text{C}$(Y) of a Baire topological space Y are characterized as those σ-homomorphisms which are induced by continuous maps from dense G₈-subsets of Y into X. This result is used to deduce a series of related results in topology and measure theory (some of which are well-known). Finally a similar result for the complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a compact Hausdorff space X tothe category algebra $\text{C}$(Y) of a Baire topological space Y is proved.     

Robinson－Ursescu Theorem in p－normed Spaces

For a convex set-valued map between p-normed (0 < p ≤ 1) spaces, we give a criterion for its inverse to be locally Lipschitz of order p. From this we obtain the Robinson-Ursescu Theorem in p-normed spaces and the open mapping and closed graph theorems for closed convex set-valued maps.     

Separable complete ANR's admitting a group structure are Hilbert manifolds

Let X be a complete-metrizable, separable ANR. The following two facts are shown: (a) if X admits a topological group structure, then either this is a Lie group structure or X is an l₂-manifold; (b) If X is a closed convex set in a complete metric linear space, then X is either locally compact or homeomorphic to l₂.     

Pairwise disjoint eight‐shaped curves in hybrid planes

We introduce a suitable notion of eight‐shaped curve in the product S × ? of a Suslin line S for the real line ?, and we prove that if S is dense in itself, then every collection of pairwise disjoint eight‐shaped curves in S × ? is countable. This parallels a folklore result which holds for the real plane. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

A relation between spaces implied by their t-equivalence

We prove that if X and Y are t-equivalent spaces (that is, if Cp(X) and Cp(Y) are homeomorphic), then there are spaces Zn, locally closed subspaces Bn of Zn, and locally closed subspaces Yn of Y, n∈N⁺, such that each Zn admits a perfect finite-to-one mapping onto a closed subspace of Xn, Yn is an image under a perfect mapping of Bn, and Y=?{Yn:n∈N⁺}. It is deduced that some classes of spaces, which for metric spaces coincide with absolute Borelian classes, are preserved by t-equivalence. Also some limitations on the complexity of spaces t-equivalent to “nice” spaces are obtained.     

Additive selections of superadditive set-valued functions

Summary A set-valued functionF from a coneC with a cone-basis of a topological vector spaceX into the family of all non-empty compact convex subsets of a locally convex spaceY is called superadditive provided thatF(x) + F(y) F(x + y), for allx, y C. We show that every superadditive set-valued function admits an additive selection.Dedicated to Professor Otto Haupt on his 100th birthday     

Vollständige und B-vollständige topologische Vektorgruppen Graphensätze und Open-Mapping Theorems

We are studying complete and B-complete topological vector groups. These Objects have been introduced by P. Kenderov [6] and D. A. Raikov [11]. They form a category TVG intermediate to the categories of topological Abelian groups and topological vector spaces and are close enough to the last one to give many useful applications to it. We first consider the problem of completion in the most used subcategories of TVG. A special functor allows to play back permanence property questions of completeness in locally convex vector groups to the same questions for locally convex vector spaces. Some examples of complete locally convex vector groups follow. We then unify some differently defined notions of B-completeness and generalize well known theorems concerning B-complete locally convex topological vector spaces to locally convex topological vector groups. Barrelledness concepts introduced in 9 and a special functor constructed in section 6 are used to formulate analogues of the closed graph and open mapping theorem for locally convex vector groups. The remainder of the note is left for applications to locally convex vector spaces. Many theorems about 1^p-sums of normed spaces are proved, as well as the B-completeness of a vast class of locally convex vector spaces including the spaces and of Köthe ([7], §13, No 5,6).