Related papers: Classification of the spaces $C_p^*(X)$ within the…

For a metrizable space $X$ of density $\kappa$, let $PM(X)$ be the space of continuous bounded pseudometrics on $X$ endowed with the uniform convergence topology. In this paper, its topology shall be classified as follows: (i) If $X$ is…

We give new characterizations of spaces $X$ which are $k_\mathbb{R}$-spaces or $s_\mathbb{R}$-spaces. Applying the obtained results we provide some sufficient and necessary conditions on $X$ for which $C_p(X)$ is a $k_\mathbb{R}$-space or…

We study expansive homeomorphisms of a compact metric space $X$ through the lens of the commutative $C^*$-algebra $C(X)$ of continuous complex-valued functions, viewed as observables of the system. We introduce the notion of expansive…

We conclude the classification of spaces of continuous functions on ordinals carried out by R. Gorak. This gives a complete topological classification of the spaces $C_p([0,\alpha])$ of all continuous real-valued functions on compact…

In our paper [18] we showed that a Tychonoff space $X$ is a $\Delta$-space (in the sense of [20], [30]) if and only if the locally convex space $C_{p}(X)$ is distinguished. Continuing this research, we investigate whether the class $\Delta$…

Let $P$ be a directed set and $X$ a space. A collection $\mathcal{C}$ of subsets of $X$ is \emph{$P$-locally finite} if $\mathcal{C}=\bigcup \{ \mathcal{C}_p : p \in P\}$ where (i) if $p \le p'$ then $\mathcal{C}_p \subseteq…

For a Tychonoff space $X$, let $C_k(X)$ and $C_p(X)$ be the spaces of real-valued continuous functions $C(X)$ on $X$ endowed with the compact-open topology and the pointwise topology, respectively. If $X$ is compact, the classic result of…

Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of projective space associated to a C*-algebra A with a fixed projection p. The resulting space P(p) admits a rich geometrical structure as a…

We determine the exact complexity of classifying compact metric spaces up to homeomorphism. More precisely, the homeomorphism relation on compact metric spaces is Borel bi-reducible with the complete orbit equivalence relation of Polish…

For a Tychonoff space $X$ by $C_p(X)$ we denote the space $C(X)$ of continuous real valued functions on $X$ endowed with the pointwise topology. We prove that an infinite compact space $X$ is scattered if and only if every closed…

We prove that each coarsely homogenous separable metric space $X$ is coarsely equivalent to one of the spaces: the sigleton, the Cantor macro-cube or the Baire macro-space. This classification is derived from coarse characterizations of the…

For a Tychonoff space $X$, we will denote by $USC_{p}(X)$ ($B_1(X)$) a set of all real-valued upper semicontinuous functions (a set of all Baire functions of class 1) defined on $X$ endowed with the pointwise convergence topology. In this…

We prove that the isomorphism relation for separable C$^*$-algebras, and also the relations of complete and $n$-isometry for operator spaces and systems, are Borel reducible to the orbit equivalence relation of a Polish group action on a…

Let X be a separable metric space and let \beta be the strict topology on the space of bounded continuous functions on X, which has the space of \tau-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonne\'{e} type…

We prove that many completeness properties coincide in metric spaces, precompact groups and dense subgroups of products of separable metric groups. We apply these results to function spaces C_p(X,G) of G-valued continuous functions on a…

We prove that in the Miller model, every $M$-separable space of the form $C_p(X)$, where $X$ is metrizable and separable, is productively $M$-separable, i.e., $C_p(X)\times Y$ is $M$-separable for every countable $M$-separable $Y$.

Let $X$ be a (real or complex) rearrangement-in\-va\-riant function space on $\Om$ (where $\Om = [0,1]$ or $\Om \subseteq \bbN$) whose norm is not proportional to the $L_2$-norm. Let $H$ be a separable Hilbert space. We characterize…

In this paper we introduce and study so-called $k^*$-metrizable spaces forming a new class of generalized metric spaces, and display various applications of such spaces in topological algebra, functional analysis, and measure theory. By…

Fix a d-minimal expansion of an ordered field. We consider the space $\mathcal D^p(M)$ of definable $\mathcal C^p$ functions defined on a definable $\mathcal C^p$ submanifold $M$ equipped with definable $\mathcal C^p$ topology. The set of…

We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if $(X,d,\mu)$ is a locally complete and separable metric measure space, then continuous functions…