Related papers: Suprema of continuous functions on connected space…

In Persistent Homology and Topology, filtrations are usually given by introducing an ordered collection of sets or a continuous function from a topological space to $\R^n$. A natural question arises, whether these approaches are equivalent…

The goal of this paper is to establish a general fixed point theorem for compact single-valued continuous mapping in Hausdorff p-vector spaces, and the fixed point theorem for upper semicontinuous set-valued mappings in Hausdorff locally…

Let $\A$ be the operator which assigns to each $m \times n$ matrix-valued function on the unit circle with entries in $H^\infty + C$ its unique superoptimal approximant in the space of bounded analytic $m \times n$ matrix-valued functions…

The famous Rosenthal-Lacey theorem asserts that for each infinite compact set $K$ the Banach space $C(K)$ admits a quotient which is either a copy of $c$ or $\ell_{2}$. What is the case when the uniform topology of $C(K)$ is replaced by the…

We investigate the following general problem, closely related to the problem of isomorphic classification of Banach spaces $C(K)$ of continuous real-valued functions on a compact space $K$, equipped with the supremum norm: Let $\mathcal{K}$…

While topology given by a linear order has been extensively studied, this cannot be said about the case when the order is given only locally. The aim of this paper is to fill this gap. We consider relation between local orderability and…

It is proved that every operator from a weak$^*$-closed subspace of $\ell_1$ into a space $C(K)$ of continuous functions on a compact Hausdorff space $K$ can be extended to an operator from $\ell_1$ to $C(K)$.

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…

A $Z$-set in a metric space $X$ is a closed subset $K$ of $X$ such that each map of the Hilbert cube $Q$ into $X$ can uniformly be approximated by maps of $Q$ into $X \setminus K$. The aim of the paper is to show that there exists a functor…

Let $M$ be the Hardy-Littlewood maximal function. Denote by $M_b$ and $[b,M]$ the maximal and the nonlinear commutators of $M$ with a function $b$. The boundedness of $M_b$ and $[b,M]$ on weighted Lebesgue spaces are characterized when the…

The aim of this paper is to consider questions concerning the possible maximum cardinality of various separable pseudoradial (in short: SP) spaces. The most intriguing question here is if there is, in ZFC, a regular (or just Hausdorff) SP…

Metric-preserving functions (here, metric aggregation functions) offer a natural method for constructing metrics on Cartesian products of metric spaces or for aggregating multiple metrics defined on a common set. Strongly metric-preserving…

We show that for every complete metric space $M$ there exists another complete metric space $N$ of the same density character such that the curve-flat quotient of $N$ is isometric to $M$. Moreover, we show that if $M$ is compact and…

Let $d$ be a metric on $R^n$ and let $C^{m,(d)}(R^n)$ be the space of $C^m$-function on $R^n$ whose partial derivatives of order $m$ belong to the space $Lip(R^n;d)$. We show that the homogeneous Sobolev space $L^{m+1}_p(R^n),p>n,$ can be…

We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, 1-increasing functions with value $1$ at $(1,1,\ldots, 1)$. We characterize the existence of a $k$-increasing…

It is solved a problem of construction of separately continuous functions on the product of compacts with a given discontinuity points set. We obtaine the following results. 1. For arbitrary \v{C}ech complete spaces $X$, $Y$ and a separable…

For a compact subset in a compact Hermitian manifold, we prove that the continuity of the extremal function at a given point in the set is a local property and that the continuity of a weighted extremal function follows from the…

We study the boundedness of commutators of the Hardy-Littlewood maximal function and the sharp maximal function on weighted Morrey spaces when the symbols of the commutators belong to weighted Lipschitz spaces. Some new characterizations…

Suppose that $f$ belongs to a suitably defined complete metric space $ {{\cal C}}^{{\alpha}}$ of H\"older $ {\alpha}$-functions defined on $[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper…

The main result of this paper states, that if a function $f:\R^2\to [0, +\infty)$ has a closed graph and the set of discontinuity points is a network (as defined by Kuratowski in Topology II, 61.IV), then the graph of $f$ is disconnected.…