Completeness,precompactness and compactness in finite-dimensional asymmetrically normed lattices

If (X,‖⋅‖)$(X,\Vert \cdot \Vert )$ is a real normed lattice, then p(x)=‖x⁺‖$p(x)=\Vert x^{+}\Vert$ defines an asymmetric norm on X. We characterise the left-K   sequentially complete, precompact and compact subsets of (R^m,p)$({\mathbb{R}}^m,p)$.     

Internal normality and internal compactness

This text contains an example which presents a way to modify any Dowker space to get a normal space X such that X×[0,1] is not κ-normal, and a theorem implying the existence of a non-Tychonoff space which is internally compact in a larger regular space. It gives answers to several questions by Arhangel'skii [A.V. Arhangel'skii, Relative normality and dense subspaces, Topology Appl. 123 (2002) 27-36].     

Expandable network and covering properties for uniform Eberlein compacta

In this paper we characterize the class of uniform Eberlein compact spaces through a network condition and also in terms of covering properties for the square of the space.     

A generalisation of the functional approach to compactness

A topological space X is compact iff the projection π:X×Y→Y is closed for any space Y. Taking this as a definition and then asking that π maps α-closed subspaces of X×Y onto β-closed subspaces of Y, for different closures α and β, extends the notion of compactness to include also examples of “asymmetric compactness” pursued in the article.Categorical closure operators and a so-called “functional approach to general topology” are employed to define and prove fundamental properties of compact objects and proper maps in this generalised setting.     

Cone monotone mappings: Continuity and differentiability

We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces, and we define two classes of monotone mappings between an ordered linear space and a metric space (resp. ordered linear space): K$K$-monotone dominated and cone-to-cone monotone mappings. K$K$-monotone dominated mappings naturally generalize mappings with finite variation (in the classical sense) and K$K$-monotone functions defined by Borwein, Burke and Lewis to mappings with domains and ranges of higher dimensions. First, using results of Veselý and Zají?ek, we show some relationships between these classes. Then, we show that every K$K$-monotone function f:X→R$f:X\to \mathbb{R}$, where X$X$ is any Banach space, is continuous outside of a set which can be covered by countably many Lipschitz hypersurfaces. This sharpens a result due to Borwein and Wang. As a consequence, we obtain a similar result for K$K$-monotone dominated and cone-to-cone monotone mappings. Finally, we prove several results concerning almost everywhere differentiability (also in metric and w^∗$w^{\ast }$-senses) of these mappings.     

Weak bases and quasi-pseudo-metrization of bispaces

In this paper it is obtained a quasi-pseudo-metrization theorem which provides a certain unification in the treatment of the biquasi-metrization problem when it is considered via sequences of neighborhoods of each point satisfying certain properties. In particular, the well-known theorems of Fox, Raghavan, Künzi, and Raghavan and Reilly are deduced from our results. We also obtain some quasi-metrization theorems in terms of pairwise locally symmetric bifunctions.     

Chain conditions and weak topologies

We study conditions on Banach spaces close to separability. We say that a topological space is pcc if every point-finite family of open subsets of the space is countable. For a Banach space E, we say that E is weakly pcc if E, equipped with the weak topology, is pcc, and we also consider a weaker property: we say that E is half-pcc if every point-finite family consisting of half-spaces of E is countable. We show that E is half-pcc if, and only if, every bounded linear map E→c₀(ω₁) has separable range. We exhibit a variety of mild conditions which imply separability of a half-pcc Banach space. For a Banach space C(K), we also consider the pcc-property of the topology of pointwise convergence, and we note that the space Cp(K) may be pcc even when C(K) fails to be weakly pcc. We note that this does not happen when K is scattered, and we provide the following example:

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There exists a non-metrizable scattered compact Hausdorff space K with C(K) weakly pcc.

     

Uniform Gateaux differentiability and weak uniform rotundity in Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm

Criteria for weak uniform rotundity and reflexivity of Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm are given. Although, criteria for uniform Gateaux differentiability and weak uniform rotundity of Musielak-Orlicz function spaces of real functions equipped with the Luxemburg norm were known, they can be also easily deduced from our main results.     

Weak separation axioms and weak covering properties

We identify some remnants of normality and call them rudimentary normality, generalize the concept of submetacompact spaces to that of a weakly subparacompact space and that of a weakly^? subparacompact space, and make a simultaneous generalization of collectionwise normality and screenability with the introduction of what is to be called collectionwise σ-normality. With these weak properties, we show that,1) on weakly subparacompact spaces, countable compactness = compactness, ω₁-compactness = Lindelöfness;2) on weakly subparacompact Hausdorff spaces with rudimentary normality, regularity = normality = countable paracompactness; and3) on weakly subparacompact regular T₁-spaces with rudimentary normality, collectionwise σ-normality = screenability = collectionwise normality = paracompactness.The famous Normal Moore Space Conjecture is thus given an even more striking appearance and Worrell and Wicke?s factorization of paracompactness (over Hausdorff spaces) along with Krajewski?s are combined and strengthened. The methodology extends itself to the factorization of paracompactness on locally compact, locally connected spaces in the manner of Gruenhage and on locally compact spaces in that of Tall, and to the factorization of subparacompactness and metacompactness in the genre of Katuta, Chaber, Junnila and Price and Smith and that of Boone, improving all of them.     

Zariski closure, completeness and compactness

The categorical theory of closure operators is used to introduce and study separated, complete and compact objects with respect to the Zariski closure operator naturally defined in any category X(A,Ω) obtained by a given complete category X (endowed with a proper factorization structure for morphisms) and by a given X-algebra (A,Ω) by forming the affine X-objects modelled by (A,Ω). Several basic examples are provided.     

Linearly ordered Radon-Nikodým compact spaces

We prove that every fragmentable linearly ordered compact space is almost totally disconnected. This combined with a result of Arvanitakis yields that every linearly ordered quasi-Radon-Nikodým compact space is Radon-Nikodým, providing a new partial answer to the problem of continuous images of Radon-Nikodým compacta.     

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.     

Subspaces of L p , p > 2, with unconditional bases have equivalent partition and weight norms

In this note we give a simple proof that every subspace of L_p, 2 < p < ∞, with an unconditional basis has an equivalent norm determined by partitions and weights. Consequently L_p has a norm determined by partitions and weights. Received: 31 January 2005     

Weak covering properties and selection principles

No convenient internal characterization of spaces that are productively Lindelöf is known. Perhaps the best general result known is Alster?s internal characterization, under the Continuum Hypothesis, of productively Lindelöf spaces which have a basis of cardinality at most _ℵ1${\aleph }_1$. It turns out that topological spaces having Alster?s property are also productively weakly Lindelöf. The weakly Lindelöf spaces form a much larger class of spaces than the Lindelöf spaces. In many instances spaces having Alster?s property satisfy a seemingly stronger version of Alster?s property and consequently are productively X, where X is a covering property stronger than the Lindelöf property. This paper examines the question: When is it the case that a space that is productively X is also productively Y, where X and Y are covering properties related to the Lindelöf property.     

A glance at spaces with closure-preserving local bases

Call a space X (weakly) Japanese at a pointx∈X if X has a closure-preserving local base (or quasi-base respectively) at the point x. The space X is (weakly) Japanese if it is (weakly) Japanese at every x∈X. We prove, in particular, that any generalized ordered space is Japanese and that the property of being (weakly) Japanese is preserved by σ-products; besides, a dyadic compact space is weakly Japanese if and only if it is metrizable. It turns out that every scattered Corson compact space is Japanese while there exist even Eberlein compact spaces which are not weakly Japanese. We show that a continuous image of a compact first countable space can fail to be weakly Japanese so the (weak) Japanese property is not preserved by perfect maps. Another interesting property of Japanese spaces is their tightness-monolithity, i.e., in every weakly Japanese space X we have for any set A⊂X.     

Weak compactness in certain star-shift invariant subspaces

The context of much of the work in this paper is that of a backward-shift invariant subspace of the form , where B is some infinite Blaschke product. We address (but do not fully answer) the question: For which B can one find a (convergent) sequence in K_B such that the sequence of real measures converges weak-star to some nontrivial singular measure on ? We show that, in order for this to hold, K_B must contain functions with nontrivial singular inner factors. And in a rather special setting, we show that this is also sufficient. Much of the paper is devoted to finding conditions (on B) that guarantee that K_B has no functions with nontrivial singular inner factors. Our primary result in this direction is based on the “geometry” of the zero set of B.     

Properties that are productive,closed-hereditary and surjective

We obtain a characterization of all those topological properties of regular Hausdorff spaces, that are preserved under the formation of arbitrary products, closed subspaces and continuous surjections.     

Hereditary normality-type properties of hyperspaces

Let X be a Hausdorff topological space and exp(X) be the space of all (nonempty) closed subsets of a space X with the Vietoris topology. We consider hereditary normality-type properties of exp(X). In particular, we prove that if exp(X) is hereditarily D-normal, then X is a metrizable compact space.     

One more topological equivalent of CH

It is noted that CH is equivalent to the assumption that every dense pseudocompact subspace of c² contains a dense Lindelöf subspace.