Related papers: Locally o-minimal structures with tame topological…

We prove that for an o-minimal expansion of the real additive group $\cal R$ and a set $P\subseteq \mathbb{R}$ of dimension $0$ such that $\langle\mathcal{R},P\rangle$ is sparse, has definable choice and every definable set has interior or…

In arXiv:1303.3724, the authors provide an axiomatic way of constructing new polynomially bounded o-minimal structures. However, all of the structures satisfying these axioms must also have smooth cell-decomposition. In this paper, we…

We introduce a notion of equivalence on tilings which is formulated in terms of their local structure. We compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are…

The present paper is devoted to questions located at the junction of the theory of space quasiconformal mappings and Riemannian surfaces. Theorems on local behavior of one class of open discrete mappings with unbounded characteristic of…

We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple group and a commutative group, and (ii) the group (G, .) is…

Many classically used function space structures (including the topology of pointwise convergence, the compact-open topology, the Isbell topology and the continuous convergence) are induced by a hyperspace structure counterpart. This scheme…

In this article we prove the topological minimality of unions of several almost orthogonal planes of arbitrary dimensions. A particular case was proved in arXiv:1103.1468, where we proved the Almgren minimality (which is a weaker property…

We establish a fixed point property for a certain class of locally compact groups, including almost connected Lie groups and compact groups of finite abelian width, which act by simplicial isometries on finite rank buildings with measurable…

A compact manifold $M$ together with a Riemannian metric $h$ on its universal cover $\tilde M$ for which $\pi_1(M)$ acts by similarities is called a similarity structure. In the case where $\pi_1(M) \not\subset \mathrm{Isom}(\tilde M, h)$…

We completely characterize definable linear orders in o-minimal structures expanding groups. For example, let (P,<_p) be a linear order definable in the real field R. Then (P,<_p) embeds definably in (R^{n+1},<_l), where <_l is the…

In this paper we present some bounds of Hausdorff measures of objects definable in o-minimal structures: sets, fibers of maps, inverse images of curves of maps, etc. Moreover, we also give some explicit bounds for semi-algebraic or…

We establish that a second countable locally compact groupoid possessing a continuous Haar system is topologically amenable if and only if it is Borel amenable. We give some examples and applications.

The class of all locally quasi-convex (lqc) abelian groups contains all locally convex vector spaces (lcs) considered as topological groups. Therefore it is natural to extend classical properties of locally convex spaces to this larger…

We explain how to see finite combinatorics of preorders implicit in the {text} of basic topological definitions or arguments in (Bourbaki, General topology, Ch.I), and define a concise combinatorial notation such that complete definitions…

We study complex non-K\"ahler manifolds with Hermitian metrics being locally conformal to metrics with special cohomological properties. In particular, we provide examples where the existence of locally conformal holomorphic-tamed…

The authors investigate the structure of quasi-o-minimal groups. Among other results, they show that quasi-o-minimal groups are abelian, that quasi-o-minimal densely ordered archimedian groups are divisible, and that every divisible…

A theorem of Elekes and Szab\'{o} recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity and dimension, definable in:…

We present characterizations of democratic property for systems of translates on a general locally compact abelian group, along a lattice in that group. That way we generalize the results from [11] on systems of integer translates.…

The subject of topological defects has become a very attractive field of study given its apparent relevance to as diverse systems as the early universe and condensed matter. As usually envisaged the topology of the manifold M of the minima…

The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss…