Related papers: On biorthogonal systems whose functionals are fini…

In [8] probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, have been used to show that for every infinite compact spaces K and L there exists a sequence $(\mu_n)$ of…

We construct a Banach space $\mathcal X_\varepsilon$ with an uncountable $\varepsilon$-biorthogonal system but no uncountable $\tau$-biorthogonal system for $\tau<\varepsilon$. In particular the space have no uncountable biorthogonal…

We prove a commutative Gelfand--Naimark type theorem, by showing that the set $C_s(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space $X$ (provided with the supremum…

The Kantorovich-Rubinshtein metric is an $L^1$-like metric on spaces of probability distributions that enjoys several serendipitous properties. It is complete separable if the underlying metric space of points is complete separable, and in…

As defined in [1], a Hausdorff space is strongly anti-Urysohn (in short: SAU) if it has at least two non-isolated points and any two infinite} closed subsets of it intersect. Our main result answers the two main questions of [1] by…

A long-standing question in the theory of measures of noncompactness is that for the Kuratowski measure of noncompactness $\alpha$ defined on a metric space $M$, and for every bounded subset $B\subset M$, is there a countable subset…

Improving a result of M. Talagrand, under the assumption of a weak form of Martin's axiom, we construct a totally disconnected compact Hausdorff space $K$ such that the Banach space $C(K)$ of continuous real-valued functions on $K$ is a…

In this paper, some features of countably $\alpha$-compact topological spaces are presented and proven. The connection between countably $\alpha$% -compact, Tychonoff, and $\alpha$-Hausdorff spaces is explained. The space is countably…

We collect several foundational results regarding the interaction between locally compact spaces, probability spaces and probability algebras, and commutative $C^*$-algebras and von Neumann algebras equipped with traces, in the…

For infinite cardinals $\kappa,\lambda$ let $C(\kappa,\lambda)$ denote the class of all compact Hausdorff spaces of weight $\kappa$ and size $\lambda$. So $C(\kappa,\lambda)=\emptyset$ if $\kappa>\lambda$ or $\lambda>2^\kappa$. If F is a…

The purpose of this paper is devoted to studying representation of measures of non generalized compactness, in particular, measures of noncompactness, of non-weak compactness, and of non-super weak compactness, etc, defined on Banach spaces…

Let $\mathcal{X}$ be a Banach space with a fundamental biorthogonal system and let $\mathcal{Y}$ be the dense subspace spanned by the vectors of the system. We prove that $\mathcal{Y}$ admits a $C^\infty$-smooth norm that locally depends on…

We construct a totally disconnected compact Hausdorff space N which has clopen subsets M included in L included in N such that N is homeomorphic to M and hence C(N) is isometric as a Banach space to C(M) but C(N) is not isomorphic to C(L).…

We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that,…

A topological space $X$ is $strongly$ $rigid$ if each non-constant continuous map $f:X\to X$ is the identity map of $X$. A Hausdorff topological space $X$ is called $Brown$ if for any nonempty open sets $U,V\subseteq X$ the intersection…

Let c=2^aleph0 denote the cardinality of the continuum and let a,b,k be infinite cardinal numbers with a<b\leq 2^a. We show that there exist precisely 2^b T0-spaces of size a and weight b up to homeomorphism. Among these non-homeomorphic…

In this paper we establish some new results concerning the Cauchy-Peano problem in Banach spaces. Firstly, we prove that if a Banach space $E$ admits a fundamental biorthogonal system, then there exists a continuous vector field $f\colon…

This paper presents a constructive proof of the existence of a regular non-atomic strictly-positive measure on any second-countable non-atomic locally compact Hausdorff space. This construction involves a sequence of finitely-additive set…

A denumerable cellular family of a topological space $\mathbf{X}$ is an infinitely countable collection of pairwise disjoint non-empty open sets of $ \mathbf{X}$. It is proved that the following statements are equivalent in $\mathbf{ZF}$:…

In the absence of the axiom of choice, new results concerning sequential, Fr\'echet-Urysohn, $k$-spaces, very $k$-spaces, Loeb and Cantor completely metrizable spaces are shown. New choice principles are introduced. Among many other…