Related papers: Total orders on compact metric spaces and covering…

We show that the order dimension of the partial order of all finite subsets of $\kappa$ under set inclusion is ${\log}_{2}({\log}_{2}(\kappa))$ whenever $\kappa$ is an infinite cardinal. We also show that the order dimension of any locally…

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…

We develop an analog to the ends of a metric space for the category of coarse metric spaces and show that it is equivalent to a previously defined coarse invariant.

Let $\mathfrak{P}$ be a topological property. We study the relation between the order structure of the set of all $\mathfrak{P}$-extensions of a completely regular space $X$ with compact remainder (partially ordered by the standard partial…

The primary objective of the present paper is to develop the theory of quantization dimension of an invariant measure associated with an iterated function system consisting of finite number of contractive infinitesimal similitudes in a…

We give an answer to the following question: for which metric in an abstract lattice the completion as a metric space coincides with the completion as a lattice. We obtain the answer for inductive limits of lattices which are complete in…

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…

In this paper, we approach the question if some of the separation axioms are equivalent in the class of asymmetric normed spaces. In particular, we make a remark on a known theorem which states that every $T_1$ asymmetric normed space with…

We prove that the Lipschitz-free space over a countable compact metric space is isometric to a dual space and has the metric approximation property.

For every countable ordinal number $\xi$, we construct a metric space $X_{\xi}$ whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both $\xi$.

In a recent paper (2018), D. Hofmann, R. Neves and P. Nora proved that the dual of the category of compact partially ordered spaces and monotone continuous maps is a quasi-variety - not finitary, but bounded by $\aleph_1$. An open question…

We generalize the notion of saturated order to infinite partial orders and give both a set-theoretic and an algebraic characterization of such orders. We then study the proof theoretic strength of the equivalence of these characterizations…

We show that there is a compact topological space carrying a measure which is not a weak* limit of finitely supported measures but is in the sequential closure of the set of such measures. We construct compact spaces with measures of…

We consider a natural way of extending the Lebesgue covering dimension to various classes of infinite dimensional separable metric spaces.

This paper concerns a study of three families of non-compact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show they have finite telescopic dimension. We also…

We begin to study classical dimension theory from the computable analysis (TTE) point of view. For computable metric spaces, several effectivisations of zero-dimensionality are shown to be equivalent. The part of this characterisation that…

We demonstrate that there is a large class of compact metric spaces for which the shadowing property can be characterized as a structural property of the space of dynamical systems. We also demonstrate for this class of spaces, that in…

We introduce and investigate a topological version of St\"ackel's 1907 characterization of finite sets, with the goal of obtaining an interesting notion that characterizes usual compactness (or a close variant of it). Define a $T_2$…

We describe the canonical correspondence between set of all finite metric spaces and set of special symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those…

The classical Cauchy completion of a metric space (by means of Cauchy sequences) as well as the completion of a uniform space (by means of Cauchy filters) are well-known to rely on the symmetry of the metric space or uniform space in…