Related papers: The quasi-metrizability of hyperspaces

We study conditions under which quasi-conformal homeomorphisms are quasi-isometries. We show that if two nilpotent geodesic Lie groups are quasi-conformally homeomorphic, then they are quasi-isometrically equivalent. We also give more…

We provide a machinery for transferring some properties of metrizable $ANR$-spaces to metrizable $LC^n$-spaces. As a result, we show that for complete metrizable spaces the properties $ALC^n$, $LC^n$ and $WLC^n$ coincide to each other. We…

Let $G$ be a group of homeomorphisms of a topological space $X$. $G$ is $\textit{(properly) isometrizable}$ if there exists a $G$-invariant (proper) gauge structure on $X$. $G$ is $\textit{equiregular}$ if for every $x \in X$ and every open…

A concept of quasi-metrizability with respect to a bornology of a generalized topological space in the sense of Delfs and Knebusch is introduced. Quasi-metrization theorems for generalized bornological universes are deduced. A uniform…

$T$-semi-selfdecomposability and subclasses $L_m(b, Q)$ and $\tilde L_m(b, Q)$ of measures on complete separable metric vector spaces are introduced and basic properties are proved. In particular, we show that $\mu$ is…

We show that $C(X)$ admits an equivalent pointwise lower semicontinuous locally uniformly rotund norm provided $X$ is Fedorchuk compact of spectral height 3. In other words $X$ admits a fully closed map $f$ onto a metric compact $Y$ such…

In this paper, we prove a theorem about embedding of some partially ordered topological spaces in topological hyperspaces equipped with Fell topology. Then we give some examples to show that the map defining the embedding may not be…

A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections.…

A topological space $Y$ is said to have (AEEP) if the following condition is fulfilled. Whenever $(X,\mathfrak{M})$ is a measurable space and $f, g: X \to Y$ are two measurable functions, then the set $\Delta(f,g) = \{x \in X:\ f(x) =…

The $H$-space, denoted as $(\mathbb{R}, \tau_{A})$, has $\mathbb{R}$ as its point set and a basis consisting of usual open interval neighborhood at points of $A$ while taking Sorgenfrey neighborhoods at points of $\mathbb{R}$-$A$. In this…

We investigate CAT(0) metric spaces whose associated Tits boundary is compact. Prominent examples of such spaces are of course the euclidean ones. However there exist non trivial geodesically complete CAT(0) spaces with compact Tits…

A non-trivial separable metric space $X$ is called an almost homology $n$-manifold if the homology groups $H_k(X,X\backslash\{x\},\mathbb Z)$ are trivial for all $x\in X$ and all $k=0,1,..,n-1$. We provide a necessary and sufficient…

We study the properties of topological spaces $(X,\tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ on $X$ has a basis that is (uniformly) definable. Examples of such spaces include the canonical…

For a metrizable space $X$, we denote by $\mathrm{Met}(X)$ the space of all metric that generate the same topology of $X$. The space $\mathrm{Met}(X)$ is equipped with the supremum distance. In this paper, for every strongly…

Let (M,g) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this…

We establish necessary and sufficient conditions guaranteeing compactness of embeddings of fractional Sobolev spaces, Besov spaces, and Triebel-Lizorkin spaces, in the general context of quasi-metric-measure spaces. Although stated in the…

We construct a metrizable semitopological semilattice $X$ whose partial order $P=\{(x,y)\in X\times X:xy=x\}$ is a non-closed dense subset of $X\times X$. As a by-product we find necessary and sufficient conditions for the existence of a…

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…

This article introduces strongly near proximity, which represents a new kind of proximity called \emph{almost proximity}. A main result in this paper is the introduction of a hit-and-miss topology on ${CL}(X)$, the hyperspace of nonempty…

A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…