Related papers: Topological groups from matrices of sets
We investigate some properties of topological groups related to disconnectedness or Archimedeanness. We prove or disprove the preservation of those under operations as subgroups, quotients, products, etc. Characterizations of…
We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and…
In this thesis we explore natural procedures through which topological structure can be constructed from specific semigroups. We will do this in two ways: 1) we equip the semigroup object itself with a topological structure, and 2) we find…
In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset…
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via…
Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…
Tree sets are posets with additional structure that generalize tree-like objects in graphs, matroids, or other combinatorial structures. They are a special class of abstract separation systems. We study infinite tree sets and how they…
We consider the problem of realizing a group as the fundamental group of a graph of groups where the vertex groups are restricted to certain classes (for example, coming from a certain finite list of groups, or having bounded geometric…
We construct the ordinary irreducible representations of the group of automorphisms of a finite rooted tree and we get a natural parametrization of them. To achieve this goals, we introduce and study the combinatorics of tree compositions,…
We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well known examples from…
It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental…
We study universal groups for right-angled buildings. Inspired by Simon Smith's work on universal groups for trees, we explicitly allow local groups that are not necessarily finite nor transitive. We discuss various topological and…
We construct a topology on a given algebraically closed field with a distinguished subfield which is also algebraically closed. This topology is finer than Zariski topology and it captures the sets definable in the pair of algebraically…
We investigate topologies on groups which arise naturally from their algebraic structure, including the Frech\'et-Markov, Hausdorff-Markov, and various kinds of Zariski topologies. Answering a question by Dikranjan and Toller, we show that…
We classify all group topologies coarser than the topology of stabilizers of finite sets in the case of automorphism groups of countable free-homogeneous structures, Urysohn space and Urysohn sphere, among other related results.
Starting from filters over the set of indices, we introduce structures in a product of sets where the coordinate sets have the given structures.
The concept of topological gyrogroups is a generalization of a topological group. In this work, ones prove that a topological gyrogroup G is metrizable iff G has an {\omega}{\omega}-base and G is Frechet-Urysohn. Moreover, in topological…
We study topological group theoretic properties of algebraic groups over local fields. In particular, we find conditions under which such groups have closed images under arbitrary continuous homomorphisms into arbitrary topological groups.
A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the…
In this article, we introduce an interesting topology-like concept concerning groups (and with almost the same method it can be defined for other algebraic systems). Given an arbitrary group $G$, we define a {\em topo-system} on $G$ as a…