Pointwise sequential compactness and weak compactness in spaces of contents

In questo lavoro si caratterizzano la compattezza sequenziale relativa e la precompattezza sequenziale (a volte detta «conditional compactness») rispetto alla topologia della convergenza puntuale per insiemi di masse aventi condominio separabile. Inoltre usando un teorema di Simons del tipo «〈x′, x _n〉→〈x′, x〉, per certix′, implica chex _n converge adx debolmente» si ottiene una relazione tra la convergenza puntuale (rispetto alla topologia debole dello spazio dei valori) e la convergenza debole per masse a codominio totalmente limitato. Questi due risultati conducono, generalizzando risultati di Graves/Ruess e Lewis, ad un criterio per la debole compattezza sequenziale relativa e per la debole precompattezza sequenziale di insiemi di masse aventi codominio separabile e totalmente limitato. L’assunzione sugli spazi (localmente convessi) dei valori dipende solo dalla dualità 〈E, E′〉; p.es. E può essere munito sia della topologia debole sia della topologia forte di spazio di Fréchet.     

Versions of separability in bitopological spaces

We study selective versions of separability in bitopological spaces. In particular, we investigate these properties in function spaces endowed with the topology of pointwise convergence and the compact-open topology.     

Fuzzy topological spaces and fuzzy compactness

It is the purpose of this paper to go somewhat deeper into the structure of fuzzy topological spaces. In doing so we found we had to alter the definition of a fuzzy topology used up to now. We shall also introduce two functors $\text{}\text{?}$gw and $\text{}\text{?}$gi which will allow us to see more clearly the connection between fuzzy topological spaces and topological spaces. Finally we shall introduce the concept of fuzzy compactness as the generalization of compactness in topology. It will be shown in a following publication that contrary to the results obtained up to now, the Tychonoff-product theorem is safeguarded with fuzzy compactness.     

Variations on selective separability in non-regular spaces

We study various variations on selective separability in non-regular topological spaces. We use the notions of θ-closure and θ-density to define selective versions of θ-separability. These properties are also related to topological games.     

Borel functions and separability of metric spaces

Let X and Y be metric spaces with X separable, and let \(f: X\rightarrow Y\) be a Borel function. Is then f(X) separable? In this paper, we prove that this problem is independent of ZFC. We also give a partial answer to an open problem which was asked by A. H. Stone.     

Almost compactness in fuzzy topological spaces

We introduce and study almost compactness for fuzzy topological spaces. We show that the almost continuous image of an almost compact fuzzy topological space is almost compact. Moreover, we show that generally almost compactness for fuzzy topological spaces is not product-invariant, but if X and Y are almost fuzzy topological spaces and X is product related to Y, then their fuzzy topological product is almost compact.     

A note on compactness in Banach spaces

Two simple concepts are presented for constructing necessary and sufficient compactness criteria in Banach spaces. The concepts are applied to the spaces lip(T), C(T), and L_p(µ), where T is a compact metric space.     

Generalized sequential normal compactness in Asplund spaces

Sequential normal compactness conditions are important properties in infinite-dimensional variational analysis and its applications. Following the recent study of the generalized sequential normal compactness (GSNC), this paper This paper reveals further applications of GSNC to the generalized differentiation theory in Asplund spaces, as well as the calculus of GSNC itself.     

Conditional weak compactness in vector-valued function spaces

Let be an ideal of over a -finite measure space and let be the Köthe dual of with . Let be a real Banach space, and the topological dual of . Let be a subspace of the space of equivalence classes of strongly measurable functions and consisting of all those for which the scalar function belongs to . For a subset of for which the set is -bounded the following statement is equivalent to conditional -compactness: the set is conditionally -compact and is a conditionally weakly compact subset of for each , with . Applications to Orlicz-Bochner spaces are given.

     

Generalized sequential normal compactness in Banach spaces

Normal compactness conditions are important properties of sets, set-valued mappings in variational analysis, which are generalized versions of the classical Lipschitzian property and are essential for the calculus of generalized differentiation theory. In this paper we propose the notion called the generalized sequential normal compactness, and establish its basic properties and calculus in general Banach spaces.     

Compactness and collective compactness in spaces of compact operators

This is a study of compactness in (a) spaces K_b(X, Y) of compact linear operators, (b) injective tensor products $\text{X}\text{}\text{?}\text{bo}{}_{\mathrm{?}}\text{Y}$, and (c) spaces L_c(X, Y) of continuous linear operators, and its various relationships with equicontinuity and collective compactness. Among the applications is a result on factoring compact sets of compact operators compactly and uniformly through one and the same reflexive Banach space.