Related papers: Dugundji's Canonical Covers, asymptotic and coveri…

We prove that if K is a remainder of the Hilbert space (i.e., K is the complement of the Hilbert space in its metrizable compactification) then every non-one-point closed image of K either contains a compact set with no transfinite…

Let $(\mathcal{X},\rho)$ be a metric space and $\lambda$ be a Borel measure on this space defined on the $\sigma$-algebra generated by open subsets of $\mathcal{X}$; this measure $\lambda$ defines volumes of Borel subsets of $\mathcal{X}$.…

It is well-known that a paracompact space $X$ is of covering dimension at most $n$ if and only if any map $f\colon X\to K$ from $X$ to a simplicial complex $K$ can be pushed into its $n$-skeleton $K^{(n)}$. We use the same idea to…

We classify compact complex surfaces which contain a Zariski open subset whose universal covering is the cylinder DxC.

We introduce the notion of spaces with weak relative cohomology and show the acyclicity of the complement $Q\setminus X$ in the Hilbert cube $Q$ of a compactum $X$ with weak relative cohomology. As a corollary we obtain the acyclicity of…

We consider the dimensions of finite type of representations of a partially ordered set, i.e. such that there is only finitely many isomorphism classes of representations of this dimension. We give a criterion for a dimension to be of…

Given a finite covering by closed convex sets of $B_X$, the unit ball of an infinite-dimensional Banach space, we investigate whether there is a set of the covering that contains balls of radius close to $1$ and (a) arbitrarily high finite…

Working in doubling metric spaces, we examine the connections between different dimensions, Whitney covers, and geometrical properties of tubular neighborhoods. In the Euclidean space, we relate these concepts to the behavior of the surface…

In this paper we prove that the asymptotic dimension of a finite-dimensional CAT(0) cube complex is bounded above by the dimension. To achieve this we prove a controlled colouring theorem for the complex. We also show that every CAT(0) cube…

A generalization of the Lebesgue number lemma is obtained. It is proved that, if each countably infinite locally finite open cover of a chainable metric space $X$ has a Lebesgue number, then $X$ is totally bounded. A property of metric…

We construct a canonical extension for strong proximity lattices in order to give an algebraic, point-free description of a finitary duality for stably compact spaces. In this setting not only morphisms, but also objects may have distinct…

This note is motivated by the article of Bamerni, Kadets and Kili\c{c}man [J. Math. Anal. Appl. 435 (2), 1812--1815 (2016)]. We consider the remaining problem which claims that if $A$ is a dense subset of a finite dimensional space $X$,…

We prove that if a quasiconvex subset $X$ of a metric space $Y$ has finite Nagata dimension and is Lipschitz $k$-connected or admits Euclidean isoperimetric inequalities up to dimension $k$ for some $k$ then $X$ is isoperimetrically…

We continue to study and present concrete examples in characteristic 2 of compound Du Val singularities defined over an algebraically closed field which have one dimensional singular loci but cannot be written as products (a rational double…

To a complex symplectic manifold X we associate a canonical quantization algebroid. Our construction is similar to that of Polesello-Schapira's deformation-quantization algebroid, but the deformation parameter is no longer central. If X is…

Finite unions of convex sets are a central object of study in discrete and computational geometry. In this paper we initiate a systematic study of complements of such unions -- i.e., sets of the form $S=\mathbb{R}^d \setminus (\cup_{i=1}^n…

For a non-compact metrizable space $X$, let ${\mathcal E}(X)$ be the set of all one-point metrizable extensions of $X$, and when $X$ is locally compact, let ${\mathcal E}_K(X)$ denote the set of all locally compact elements of ${\mathcal…

We show that a space with a finite asymptotic dimension is embeddable in a non-positively curved manifold. Then we prove that if a uniformly contractible manifold X is uniformly embeddable in $\R^n$ or non-positively curved n-dimensional…

In this paper we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form $$ \kappa({\mathcal U}) \le…

Canonical extension of finitary ordered structures such as lattices, posets, proximity lattices, etc., is a certain completion which entirely describes the topological dual of the ordered structure and it does so in a purely algebraic and…