Related papers: On weakly Radon-Nikod\'ym compact spaces

We prove that a continuous image of a Radon-Nikod\'ym compact space of weight less than b is Radon-Nikod\'ym compact. As a Banach space counterpart, subspaces of Asplund generated Banach spaces of density character less than b are Asplund…

A compact space is said to be weakly Radon-Nikod\'{y}m (WRN) if it can be weak*-embedded into the dual of a Banach space not containing $\ell_1$. We investigate WRN Boolean algebras, i.e. algebras whose Stone space is WRN compact. We show…

Let X be a separable Banach space and Y a space which has the Radon-Nikodym property. In this work, we show that L(X, Y) has the Radon-Nikodym property, if L(X, Y) is weakly locally uniformly convex or if L(X, Y) is a weakly compactly gen-…

We provide a characterization of continuous images of Radon-Nikodym compacta lying in a product of real lines and model on it a method for constructing natural examples of such continuous images.

We prove in this short report that for arbitrary weak converging sequence of sigma-finite Borelian measures in the separable Banach space there is a compact embedded separable subspace such that this measures not only are concentrated in…

We study classes of Banach spaces where the points of weak$^*$-weak continuity for the identity mapping on the dual unit ball form a weak$^*$-dense and weak$^*$-$G_{\delta}$ set. We also discuss how this property behaves in higher duals of…

A compact space $K$ is Radon-Nikod\'{y}m if there is a lower semi-continuous metric fragmenting $K$. In this note, we show that, under $\diamondsuit (\mathrm{non}{\mathcal{M}})$, there is a Radon-Nikod\'{y}m compact space of weight…

The notion of super weak compactness for subsets of Banach spaces is a strengthening of the weak compactness that can be described as a local version of super-reflexivity. A recent result of K. Tu which establishes that the closed convex…

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-Nikodym compact space is Radon-Nikodym, providing a…

We construct a continuous image of a Radon-Nikod\'ym compact space which is not Radon-Nikod\'ym compact, solving the problem posed in the 80ties by Isaac Namioka.

Let $E$ be a Banach space such that $E'$ has the Radon-Nikod\'ym property. The aim of this work is to connect relative weak compactness in the $E$-valued martingale Hardy space $H^{1}(\mu,E)$ to a convex compactness criterion in a weaker…

Banach spaces without the Radon-Nikod\'ym property are characterized as spaces containing bilipschitz images of thick families of geodesics defined as follows. A family $T$ of geodesics joining points $u$ and $v$ in a metric space is called…

The weak Whyburn property is a generalization of the classical sequential property that has been studied by many authors. A space $X$ is weakly Whyburn if for every non-closed set $A \subset X$ there is a subset $B \subset A$ such that…

In this paper we show that if $(y_n)$ is a seminormalized sequence in a Banach space which does not have any weakly convergent subsequence, then it contains a wide-$(s)$ subsequence $(x_n)$ which admits an equivalent convex basic sequence.…

Assuming that there is a stationary set in $\omega_{2}$ of ordinals of countable cofinality that does not reflect, we prove that there exists a compact space which is not Corson compact and whose all continuous images of weight at most…

The paper presents some weak compactness criterion for a subset $M$ of the set $\mathfrak{RM}_b(T,\mathcal{G})$ of all positive bounded Radon measures on a Hausdorff topological space $(T,\mathcal{G})$ similar to the Prokhorov criterion for…

In this paper the weak topology on a normed space is studied from the viewpoint of infinite-dimensional topology. Besides the weak topology on a normed space $X$ (coinciding with the topology of uniform convergence on finite subsets of the…

For an arbitrary topological group G any compact G-dynamical system (G,X) can be linearly G-represented as a weak*-compact subset of a dual Banach space V*. As was shown by Megrelishvili (2003), the Banach space V can be chosen to be…

We list a number of problems in several topics related to compactness in nonseparable Banach spaces. Namely, about the Hilbertian ball in its weak topology, spaces of continuous functions on Eberlein compacta, WCG Banach spaces, Valdivia…

A topological space is said to be sequential if every sequentially closed subspace is closed. We consider Banach spaces with weak*-sequential dual ball. In particular, we show that if $X$ is a Banach space with weak*-sequentially compact…