Conway’s Question: The Chase for Completeness

We study various degrees of completeness for a Tychonoff space X. One of them plays a central role, namely X is called a Conway space if X is sequentially closed in its Stone–?ech compactification β X (a prominent example of Conway spaces is provided by Dieudonné complete spaces). The Conway spaces constitute a bireflective subcategory Conw of the category Tych of Tychonoff spaces. Replacing sequential closure by the general notion of a closure operator C, we introduce analogously the subcategory Conw _C of C-Conway spaces, that turns out to be again a bireflective subcategory of Tych. We show that every bireflective subcategory of Tych can be presented in this way by building a Galois connection between bireflective subcategories of Tych and closure operators of Top finer than the Kuratowski closure. Other levels of completeness are considered for the (underlying topological spaces of) topological groups. A topological group G is sequentially complete if it is sequentially closed in its Ra?kov completion ${ \ifmmode\expandafter\tilde\else\expandafter\~\fi{G}}$ . The sequential completeness for topological groups is stronger than Conway’s property, although they coincide in some classes of topological groups, for example: free (Abelian) topological groups, pseudocompact groups, etc.     

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 unique exchange property for bases

We define the concept of unique exchange on a sequence (X₁,…, X_m) of bases of a matroid M as an exchange of x ? X_i for y ? X_j such that y is the unique element of X_j which may be exchanged for x so that (X_i ? {x}) ∪ {y} and (X_j ? {y}) ∪ {x} are both bases. Two sequences X and Y are compatible if they are on the same multiset. Let UE(1) [UE(2)] denote the class of matroids such that every pair of compatible basis sequences X and Y are related by a sequence of unique exchanges [unique exchanges and permutations in the order of the bases]. We similarly define UE(3) by allowing unique subset exchanges. Then UE(1),UE(2), and UE(3) are hereditary classes (closed under minors) and are self-dual (closed under orthogonality). UE(1) equals the class of series-parallel networks, and UE(2) and UE(3) are contained in the class of binary matroids. We conjecture that UE(2) contains the class of unimodular matroids, and prove a related partial result for graphic matroids. We also study related classes of matroids satisfying transitive exchange, in order to gain information about excluded minors of UE(2) and UE(3). A number of unsolved problems are mentioned.     

Function spaces and one point extensions for the construct of metered spaces

The construct M of metered spaces and contractions is known to be a superconstruct in which all metrically generated constructs can be fully embedded. We show that M has one point extensions and that quotients in M are productive. We construct a Cartesian closed topological extension of M and characterize the canonical function spaces with underlying sets Hom(X,Y) for metered spaces X and Y. Finally we obtain an internal characterization of the objects in the Cartesian closed topological hull of M.     

Various disconnectivities of spaces and projectabilities of ?-groups

Arch denotes the category of archimedean ?-groups and ?-homomorphisms. Tych denotes the category of Tychonoff spaces with continuous maps, and α denotes an infinite cardinal or ∞. This work introduces the concept of an αcc-disconnected space and demonstrates that the class of αcc-disconnected spaces forms a covering class in Tych. On the algebraic side, we introduce the concept of an αcc-projectable ?-group and demonstrate that the class of αcc-projectable ?-groups forms a hull class in Arch. In addition, we characterize the αcc-projectable objects in W—the category of Arch-objects with designated weak unit and ?-homomorphisms that preserve the weak unit—and construct the αcc-hull for G in W. Lastly, we apply our results to negatively answer the question of whether every hull class (resp., covering class) is epireflective (resp., monocoreflective) in the category of W-objects with complete ?-homomorphisms (resp., the category of compact Hausdorff spaces with skeletal maps).     

The Topology of Limits of Direct Systems Induced by Maps

Let Z, H be spaces. In previous work, we introduced the direct system X induced by the set of maps between the spaces Z and H. Now we will consider the case that X is induced by possibly a proper subset of the maps of Z to H. Our objective is to explore conditions under which X = dirlim X will be T₁, Hausdorff, regular, completely regular, pseudo-compact, normal, an absolute co-extensor for some space K, or will enjoy some combination of these properties.     

On unconditional polynomial bases inL p and bergman spaces

In this paper we consider unconditional bases inL _p(T), 1<p<∞,p ≠ 2, consisting of trigonometric polynomials. We give a lower bound for the degree of polynomials in such a basis (Theorem 3.4) and show that this estimate is best possible. This is applied to the Littlewood-Paley-type decompositions. We show that such a decomposition has to contain exponential gaps. We also consider unconditional polynomial bases inH _p as bases in Bergman-type spaces and show that they provide explicit isomorphisms between Bergman-type spaces and natural sequences spaces.     

A Note on Kato–Nakayama spaces

In this Note we prove the following result. A fine log scheme over the complex numbers and its saturated have homeomorphic Kato–Nakayama associated spaces. Moreover these spaces are isomorphic as ringed spaces, either with the ring sheaf defined by Kato–Nakayama, or with that defined by Ogus. In the definition of these spaces, non-integral monoids are involved, so that the proof of the result is based on properties of nonnecessarily integral monoids. To cite this article: M. Cailotto, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On a residual local projection method for the Darcy equation

A new symmetric local projection method built on residual bases (RELP) makes linear equal-order finite element pairs stable for the Darcy problem. The derivation is performed inside a Petrov–Galerkin enriching space approach (PGEM) which indicates parameter-free terms to be added to the Galerkin method without compromising consistency. Velocity and pressure spaces are augmented using solutions of residual dependent local Darcy problems obtained after a static condensation procedure. We prove the method achieves error optimality and indicates a way to recover a locally mass conservative velocity field. Numerical experiments validate theory. To cite this article: L.P. Franca et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On bounded multilinear forms on a class of lp spaces

In this paper we generalize an old result of Littlewood and Hardy about bilinear forms defined in a class of sequence spaces. Historically, Littlewood [Quart. J. Math.1 (1930)] first proved a result on bilinear forms on bounded sequences and this result was then generalized by Hardy and Littlewood in a joint paper [Quart. J. Math.5(1934)] to bilinear forms on a class of l^p spaces. Later Davie and Kaijser proved Littlewood's results for multilinear forms. In this paper, Theorems A and B generalize the results to multilinear forms on l^p spaces. All the results are stated at the end of Section 1. Theorems A and B are proved, respectively, in Sections 2 and 3.     

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.     

Bracketing Metric Entropy Rates and Empirical Central Limit Theorems for Function Classes of Besov- and Sobolev-Type

We derive ? ^r (μ)-bracketing metric and sup-norm metric entropy rates of bounded subsets of general function spaces defined over ? ^d or, more generally, over Borel subsets thereof, by adapting results of Haroske and Triebel (Math. Nachr. 167, 131–156, 1994; 278, 108–132, 2005). The function spaces covered are of (weighted) Besov, Sobolev, Hölder, and Triebel type. Applications to the theory of empirical processes are discussed. In particular, we show that (norm-)bounded subsets of the above mentioned spaces are Donsker classes uniformly in various sets of probability measures.     

The concept lattice functors

This paper is concerned with the relationship between contexts, closure spaces, and complete lattices. It is shown that, for a unital quantale L, both formal concept lattices and property oriented concept lattices are functorial from the category L-Ctx of L-contexts and infomorphisms to the category L-Sup of complete L-lattices and suprema-preserving maps. Moreover, the formal concept lattice functor can be written as the composition of a right adjoint functor from L-Ctx to the category L-Cls of L-closure spaces and continuous functions and a left adjoint functor from L-Cls to L-Sup.     

Stability constants and the homology of quasi-Banach spaces

We affirmatively solve the main problems posed by Laczkovich and Paulin in Stability constants in linear spaces, Constructive Approximation 34 (2011), 89–106 (do there exist cases in which the second Whitney constant is finite while the approximation constant is infinite?) and by Cabello and Castillo in The long homology sequence for quasi-Banach spaces, with applications, Positivity 8 (2004), 379–394 (do there exist Banach spaces X, Y for which Ext(X, Y) is Hausdorff and nonzero?). In fact, we show that these two problems are the same.     

Some curvature properties of Cartan spaces with mth root metrics

In this paper, we define some non-Riemannian curvature properties for Cartan spaces. We consider a Cartan space with the mth root metric. We prove that every mth root Cartan space of isotropic Landsberg curvature, or isotropic mean Landsberg curvature, or isotropic mean Berwald curvature reduces to a Landsberg, weakly Landsberg, and weakly Berwald spaces, respectively. Then we show that the mth root Cartan space of almost vanishing H-curvature satisfies H?=?0.     

Lorentz Spaces and Embeddings Induced by Almost Greedy Bases in Banach Spaces

The aim of this paper is to undertake a systematic qualitative study of the built-in symmetry of almost greedy bases in Banach spaces. More specifically, by refining the techniques that Wojtaszczyk used in J Approx Theory 107(2), 293–314 2000 for quasi-greedy bases in Hilbert spaces, we show that an almost greedy basis in a Banach space X naturally induces embeddings that allow sandwiching X between two symmetric sequence spaces. Using classical interpolation techniques in combination with duality, we also explore what we label as interpolation of greedy bases. It is then proved that the only almost greedy basis shared by any two \(\ell _{p}\) spaces is equivalent to the standard unit vector basis and that there is no basis which is simultaneously (normalized and) greedy in two different \(L_{p}\) spaces. As a by-product of our work, we obtain a new characterization of greedy bases in Banach spaces in terms of bounded linear operators.     

Complex interpolation spaces on multiply-connected domains

Variants of Calderón's interpolation spaces [A₀, A₁]_θ which are defined using a multiply-connected domain instead of the strip 0 < Re z < 1 are considered. It is shown that they coincide to within equivalence of norms with Calderón's spaces. This result applies also to spaces obtained by generalised forms of Calderón's construction due to Coifman, Cwikel, Rochberg, Sagher, and Weiss (Advan. in Math.43 (1982), 203–229).     

On the application of fibred mapping spaces to exponential laws for bundles,ex-spaces and other categories of maps

A previous paper constructed exponential laws in the category Top_B of spaces over B. The present paper relates these laws to constructions known for locally trivial maps, and constructs also new exponential laws for ex-spaces, fibred section spaces and fibred relative lifting spaces. Versions of these laws for homotopy classes of maps are discussed.     

The main invariants of a complex finsler space

In this paper we extend the results obtained in [3], where are investigated the general settings of the two-dimensional complex Finsler manifolds, with respect to a local complex Berwald frame. The geometry of such manifolds is controlled by three real invariants which live on T'M: two horizontal curvature invariants K and W and one vertical curvature invariant I. By means of these invariants are defined both the horizontal and the vertical holomorphic sectional curvatures. The complex Landsberg and Berwald spaces are of particular interest. Complex Berwald spaces coincide with Kähler spaces, in the two – dimensional case. We establish the necessary and sufficient condition under which K is a constant and we obtain a characterization for the Kähler purely Hermitian spaces by the fact K = W = constant and I = 0. For the class of complex Berwald spaces we have K = W = 0. Finally, a classification of two-dimensional complex Finsler spaces for which the horizontal curvature satisfies a special property is obtained.