We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos’s property ${\alpha }_{1.5}$ is equivalent to Arhangel’skiĭ’s formally stronger property α₁. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space X such that the space $C_p\left(X\right)$ of continuous real-valued functions on X with the...

We consider the hyperspace $CL\left(X\right)$ of nonempty closed subsets of completely metrizable space $X$ endowed with the Wijsman topologies ${\tau }_{W_d}$. If $X$ is separable and $d$, $e$ are two metrics generating the topology of $X$, every countable set closed in $\left(CL\left(X\right),{\tau }_{W_e}\right)$ has isolated points in $\left(CL\left(X\right),{\tau }_{W_d}\right)$. For $d=e$ , this implies a theorem of Costantini on topological completeness of $\left(CL\left(X\right),{\tau }_{W_d}\right)$. We show that for nonseparable $X$ the hyperspace $\left(CL\left(X\right),{\tau }_{W_d}\right)$ may contain a closed copy of the rationals. This answers a question of Zsilinszky.

Let $\mathrm{\Omega }$ be a bounded open subset of $R^n$, $n>4$, of class $C^2$ . Let $u\in H^2\left(\mathrm{\Omega }\right)$ a solution of elliptic non linear non variational system $$a\left(x,u,Du,H\left(u\right)\right)=b\left(x,u,Du\right)$$ where $a\left(x,u,\mu ,\xi \right)$ and $b\left(x,u,\mu \right)$ are vectors in $R^N$, $N\ge 1$, measurable in $x$, continuous in $\left(u,\mu ,\xi \right)$ and $\left(u,\mu \right)$ respectively. Here, we demonstrate that if $b\left(x,u,\mu \right)$ has limit controlled growth, if $a\left(x,u,\mu ,\xi \right)$ is of class $C^1$ in $\xi$ and satisfies the Campanato condition $\left(A\right)$ and, together with $\frac{\partial a}{\partial \xi }$, certain continuity assumptions, then the vector $Du$ is partially Hölder continuous for every exponent $\alpha <1-\frac{n}{p}$.

Corresponding to [27], some properties of S-closed subspaces and subsets $𝒮$-closed relative to a topological space are proved. Conditions under which mappings preserve certain $𝒮$-closed subspaces are investigated.

We study relations between the cellularity and index of narrowness in topological groups and their $G_{\delta }$-modifications. We show, in particular, that the inequalities $in\left({\left(H\right)}_{\tau }\right)\le 2^{\tau ·in\left(H\right)}$ and $c\left({\left(H\right)}_{\tau }\right)\le 2^{2^{\tau ·in\left(H\right)}}$ hold for every topological group $H$ and every cardinal $\tau \ge \omega$, where ${\left(H\right)}_{\tau }$ denotes the underlying group $H$ endowed with the $G_{\tau }$-modification of the original topology of $H$ and $in\left(H\right)$ is the index of narrowness of the group $H$. Also, we find some bounds for the complexity of continuous real-valued functions $f$ on an arbitrary $\omega$-narrow group...

Questo articolo considera una successione di equazioni differenziali a derivate parziali non lineari in forma di divergenza del tipo $$-\text{div}\left(Q^{ϵ}G\left(x,N^{ϵ}\nabla u\right)\right)=f^{ϵ},$$ in un dominio limitato $\mathrm{\Omega }$ dello spazio $n$-dimensionale; $Q^{ϵ}=Q^{ϵ}\left(x\right)$ e $N^{ϵ}=N^{ϵ}\left(x\right)$ sono matrici con coefficenti limitati, $N^{ϵ}$ e è invertibile e la sua matrice inversa $R^{ϵ}$ ha anche coefficenti limitati. La non linearità è dovuta alla funzione $G=G\left(x,\xi \right)$; la condizione di crescita, la monotonicità e le ipotesi di coercitività sono modellate sul $p$-Laplaciano, $1<p<\mathrm{\infty }$, ed assicurano l'esistenza di...

A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, $U_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}$, such that $U_{\alpha }\subset U_{\beta }$ whenever β ≤ α for all $\alpha ,\beta \in {\mathbb{N}}^{\mathbb{N}}$. The class of all metrizable topological groups is a proper subclass of the class $T{G}_{}$ of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group $G\in T{G}_{}$ is metrizable, and hence G is strictly angelic. We deduce from...