Related papers: Some results on separate and joint continuity

For a set $X\sbst\R$, let $B(X)\sbst\R^X$ denote the space of Borel real-valued functions on $X$, with the topology inherited from the Tychonoff product $\R^X$. Assume that for each countable $A\sbst B(X)$, each $f$ in the closure of $A$ is…

For every monic polynomial $f \in \mathbb{Z}[X]$ with $\operatorname{deg}(f) \geq 1$, let $\mathcal{L}(f)$ be the set of all linear recurrences with values in $\mathbb{Z}$ and characteristic polynomial $f$, and let \begin{equation*}…

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…

Let C(K) denote the Banach algebra of continuous real functions, with the supremum norm, on a compact Hausdorff space K. For two subsets of C(K), one can define their product by pointwise multiplication, just as the Minkowski sum of the…

Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k term arithmetic progressions and such collection is also piecewise syndetic in Z: They used algebraic structure of beta N. The above result…

The main result of this paper is the proof of the simultaneous consistency, modulo a weakly compact cardinal, of the equality $2^{< \mathfrak{c}} = \mathfrak{c}$ with the following property (*) of partitions of pairs of $\mathfrak{c}$:…

We show that the Continuum Hypothesis is consistent with all regular spaces of hereditarily countable $\pi$-character being C-closed. This gives us a model of ZFC in which the Continuum Hypothesis holds and compact Hausdorff spaces of…

A topological space $X$ is said to be {\em $Y$-rigid} if any continuous map $f:X\rightarrow Y$ is constant. In this paper we construct a number of examples of regular countably compact $\mathbb R$-rigid spaces with additional properties…

We show that for any locally compact second countable group $G$ and any continuous positive definite function $\phi:G\rightarrow\mathbb{C}$, there exists an ergodic measure preserving system $(X,\mathscr{B},\mu,\{T_g\}_{g \in G})$ and a…

We prove that every place P of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension E of F. We show that E|F can be chosen to be Galois, after a finite purely inseparable extension of…

Let $C({\mathbb R}^n)$ denote the set of real valued continuous functions defined on ${\mathbb R}^n$. We prove that for every $n\ge 2$ there are positive numbers $\lambda _1 , \ldots , \lambda _n$ and continuous functions $\phi_1 ,\ldots ,…

Let $K$ be a field of characteristic $0$, and let $k \geq 2$ be an integer. We prove that every $K$-linear bijection $f \colon K[X] \to K[X]$ strongly preserving the set of $k$-free polynomials (or the set of polynomials with a $k$-fold…

The problem of the existence of non-pseudo-$\aleph_1$-compact $\mathbb R$-factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than $\omega_1$. Closely related results concerning the…

It is known that $C(X)$ is algebraically closed if $X$ is a locally connected, hereditarily unicoherent compact Hausdorff space. For such spaces, we prove that if $F:C(X) \to C(X)$ is given by an everywhere convergent power series with…

We prove the following results. 1. If $X$ is a $\alpha$-favourable space, $Y$ is a regular space, in which every separable closed set is compact, and $f:X\times Y\to\mathbb R$ is a separately continuous everywhere jointly discontinuous…

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…

We consider the space $\mathcal{D}'^r_L(M;E)$ of distributional sections of the smooth complex vector bundle $E\rightarrow M$ whose Sobolev wave front set of order $r\in\mathbb{R}$ lies in the closed conic subset $L$ of $T^*M\backslash0$.…

A space $X$ has countable $(F)$-property if it has countable point network satisfying the Collins-Roscoe structuring mechanism. Some sufficient conditions for $C_p(X)$ having countable $(F)$-property are obtained. As a corollary, we prove…

We prove that the smoothed counting function of the set of quartic fields, satisfying any finite set of local conditions, can be written as a linear combination of $X,X^{5/6}\log X,X^{5/6}$, upto an error term of $O(X^{13/16+o(1)})$. For…

Let M be the countably infinite metric fan. We show that C_k(M,2) is sequential and contains a closed copy of Arens space S_2. It follows that if X is metrizable but not locally compact, then C_k(X) contains a closed copy of S_2, and hence…