Related papers: On the Semicontinuity in Product Spaces

An equation $f(x)=a$, where $f$ is a complex meromorphic function and $a\in\mathbb{C}$ is a parameter, is solvable in elementary functions if the inverse map $x=f^{-1}(a)$ can be expressed as a finite composition of arithmetic operations…

We consider two natural topologies on the space $S(X\times Y,Z)$ of all separately continuous functions defined on the product of two topological spaces $X$ and $Y$ and ranged into a topological or metric space $X$. These topologies are the…

We prove that a real x is 1-generic if and only if every differentiable computable function has continuous derivative at x. This provides a counterpart to recent results connecting effective notions of randomness with differentiability. We…

Furstenberg, Katznelson and Weiss proved in the early 1980s that every measurable subset of the plane with positive density at infinity has the property that all sufficiently large real numbers are realised as the Euclidean distance between…

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…

Let $X$ be a topological vector space of complex-valued sequences and $Y$ be a subset of $X$. We provide conditions for $X \setminus Y \cup \{0\}$ to contain uncountably infinitely many linearly independent dense vector subspaces of $X$. We…

We study the existence of continuity points for mappings $f: X\times Y\to Z$ whose $x$-sections $Y\ni y\to f(x,y)\in Z$ are fragmentable and $y$-sections $X\ni x\to f(x,y)\in Z$ are quasicontinuous, where $X$ is a Baire space and $Z$ is a…

The aim of this article is to define a notion of cardinal utility function called measurable utility and to define it on a connected and separable subset of a weakly ordered topological space. The definition is equivalent to the ones given…

Let h be a real-analytic function in the neighborhood of some compact set K on the plane. We show that for any complex measure on the Euclidean space of a finite total variation without singular components with the Fourier--Stieltjes…

Let $G$ be an infinite countable amenable group and let $(X,G)$ be a $G$-subshift with specification, containing a free element. We prove that $(X,G)$ is universal, i.e., has positive topological entropy and for any free ergodic $G$-action…

We say that a metrizable space $M$ is a Krasinkiewicz space if any map from a metrizable compactum $X$ into $M$ can be approximated by Krasinkiewicz maps (a map $g\colon X\to M$ is Krasinkiewicz provided every continuum in $X$ is either…

In this article, we discuss the semicontinuity problem of certain properties on fibers for a morphism of schemes. One aspect of this problem is local. Namely, we consider properties of schemes at the level of local rings, in which the main…

We obtain necessary conditions for the existence of complete vertical graphs of constant mean curvature in the Hyperbolic and Steady State spaces. In the two-dimensional case we prove Bernstein-type results in each of these ambient spaces.

A landmark theorem in the metric theory of continued fractions begins this way: Select a non-negative real function $f$ defined on the positive integers and a real number $x$, and form the partial sums $s_n$ of $f$ evaluated at the partial…

Let (X,m) and (Y,n) be standard measure spaces. A function f in $L^\infty(X\times Y,m\times n)$ is called a (measurable) Schur multiplier if the map $S_f$, defined on the space of Hilbert-Schmidt operators from $L_2(X,m)$ to $L_2(Y,n)$ by…

Let $X$ and $Y$ be topological spaces. Let $C$ be a path-connected closed set of $X\times Y$. Suppose that $C$ is locally direct product, that is, for any $(a,b)\in X\times Y$, there exist an open set $U$ of $X$, an open set $V$ of $Y$, a…

It is shown that a set in product of $n$ metrizable spaces is the discontinuity points set of some separately continuous function if and only if this set can be represented as the union of a sequence of $F_{\sigma}$-sets which are locally…

We prove that the Reeb space of a proper definable map $f:X \rightarrow Y$ in an arbitrary o-minimal expansion of a real closed field is realizable as a proper definable quotient. This result can be seen as an o-minimal analog of Stein…

We introduce partial secondary invariants associated to complete Riemannian metrics which have uniformly positive scalar curvature outside a prescribed subset on a spin manifold. These can be used to distinguish such Riemannian metrics up…

Let ${\mathbb T}=({\bf T},\leq)$ and ${\mathbb T}_{1}=({\bf T}_{1},\leq_{1})$ be linearly ordered sets and $\mathscr{X}$ be a topological space. The main result of the paper is the following: If function $\boldsymbol{f}(t,x):{\bf…