Related papers: On the problem of compact totally disconnected ref…

We work in set-theory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak…

Let $Z$ be a fixed separable operator space, $X\subset Y$ general separable operator spaces, and $T:X\to Z$ a completely bounded map. $Z$ is said to have the Complete Separable Extension Property (CSEP) if every such map admits a completely…

Given a category of objects, it is both useful and important to know if all the objects in the category may be realised as sub-objects -- via morphisms in the given category -- of a single object in that category enjoying some nice…

By a recent result of Juh\'{a}sz and van Mill, a locally compact topological group whose dense subspaces are all separable is metrizable. In this note we investigate the following question: is every locally compact group having all dense…

We say that a metric space $(X,d)$ possesses the \emph{Banach Fixed Point Property (BFPP)} if every contraction $f:X\to X$ has a fixed point. The Banach Fixed Point Theorem says that every complete metric space has the BFPP. However, E.…

The KC property, a separation axiom between weakly Hausdorff and Hausdorff, requires compact subsets to be closed. Various assumptions involving local conditions, dimension, connectivity, and homotopy show certain KC-spaces are in fact…

Recently there has been interest in pairs of Banach spaces $(E_0,E)$ in an $o-O$ relation and with $E_0^{**}=E$. It is known that this can be done for Lipschitz spaces on suitable metric spaces. In this paper we consider the case of a…

We construct a compact Hausdorff space $K$ such that the space $P(K)$ of Radon probabiblity measures on $K$ considered with the weak$^*$ topology (induced from the space of continuous functions $C(K)$) is countably tight which is a…

The famous Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for…

For two not necessarily commutative topological groups G and T, let H(G,T) denote the space of all continuous homomorphisms from G to T with the compact-open topology. We prove that if G is metrizable and T is compact then H(G,T) is a…

We prove that each non-metrizable sequential rectifiable space $X$ of countable $cs^*$-character contains a clopen rectifiable submetrizable $k_\omega$-subspace $H$ and admits an open disjoint cover by subspaces homeomorphic to clopen…

Let $X$ be a separable nonquasireflexive Banach space. Let $Y$ be a Banach space isomorphic to a subspace of $X^*$. The paper is devoted to the following questions: 1. Under what conditions does there exist an isomorphic embedding $T:Y\to…

We show that, given a Banach space $X$, the Lipschitz-free space over $X$, denoted by $\mathcal{F}(X)$, is isomorphic to $(\sum_{n=1}^\infty \mathcal{F}(X))_{\ell_1}$. Some applications are presented, including a non-linear version of…

Let $\Gamma$ be a Polish space and let $K$ be a separable and pointwise compact set of real-valued functions on $\Gamma$. It is shown that if each function in $K$ has only countably many discontinuities then $C(K)$ may be equipped with a…

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 study subspaces of Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,\dots$, of a function $f\in L_M$, $\int_0^1 f(t)\,dt=0$. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. In terms of…

By a metric fractal we understand a compact metric space $K$ endowed with a finite family $\mathcal F$ of contracting self-maps of $K$ such that $K=\bigcup_{f\in\mathcal F}f(K)$. If $K$ is a subset of a metric space $X$ and each…

A topological space is almost locally compact if it contains a dense locally compact subspace. We generalize a result from \cite{Ma}, showing that isomorphism on Borel classes of almost locally compact Polish metric structures is always…

We establish a connection between continuous K-theory and integral cohomology of rigid spaces. Given a rigid analytic space over a complete discretely valued field, its continuous K-groups vanish in degrees below the negative of the…

The complemented subspace problem asks, in general, which closed subspaces $M$ of a Banach space $X$ are complemented; i.e. there exists a closed subspace $N$ of $X$ such that $X=M\oplus N$? This problem is in the heart of the theory of…