• Corpus ID: 118558800


  author={N. Brodskiy and Jerzy Dydak and B. Labuz and A. Mitra},
  journal={arXiv: Algebraic Topology},
We discuss various uniform structures and topologies on the uni- versal covering space e X and on the fundamental group �1(X, x0). We intro- duce a canonical uniform structure CU(X) on a topological space X and use it to relate topologies on e X and uniform structures on ^ CU(X). Using our concept of universal Peano space we show connections between the topology introduced by Spanier (30) and a uniform structure of Berestovskii and Plaut (2). We give a sufficient and necessary condition… 
On Topologized Fundamental Groups and Covering Groups of Topological Groups
We show that every topological group is a strong small loop transfer space at the identity element. This implies that for a connected locally path connected topological group G, the universal path
On Generalized Covering Groups of Topological Groups
It is well-known that a homomorphism p between topological groups K, G is a covering homomorphism if and only if p is an open epimorphism with discrete kernel. In this paper we generalize this fact,
Generalized uniform covering maps relative to subgroups
Abstract In “Rips complexes and covers in the uniform category” (Brodskiy et al., preprint [4] ) the authors define, following James (1990) [5] , covering maps of uniform spaces and introduce the
On Topologized Fundamental Group and covering spaces of topological groups
In this paper, we show that every topological group is a strong small loop transfer space at the identity element. This implies that the quasitopological fundamental group of a connected locally path
1-Dimensional intrinsic persistence of geodesic spaces
  • Žiga Virk
  • Mathematics
    Journal of Topology and Analysis
  • 2018
Given a compact geodesic space [Formula: see text], we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of [Formula: see
Fibrations, unique path lifting, and continuous monodromy
Abstract Given a path-connected space X and H ≤ π 1 ( X , x 0 ) , there is essentially only one construction of a map p H : ( X ˜ H , x ˜ 0 ) → ( X , x 0 ) with connected and locally path-connected
On Topologized Fundamental Groups with Small Loop Transfer Viewpoints
In this paper, by introducing some kind of small loop transfer spaces at a point, we study the behavior of topologized fundamental groups with the compact-open topology and the whisker topology,
On strong small loop transfer spaces relative to subgroups of fundamental groups
Let $H$ be a subgroup of the fundamental group $\pi_{1}(X,x_{0})$. By extending the concept of strong SLT space to a relative version with respect to $H$, strong $H$-SLT space, first, we investigate
On Subgroups of Topologized Fundamental Groups and Generalized Coverings
In this paper, we are interested in study subgroups of topologized fundamental groups and their influences on generalized covering maps. More precisely, we find some relationships between generalized
On Topological Homotopy Groups and Relation to Hawaiian Groups
By generalizing the whisker topology on the $n$th homotopy group of pointed space $(X, x_0)$, denoted by $\pi_n^{wh}(X, x_0)$, we show that $\pi_n^{wh}(X, x_0)$ is a topological group if $n \ge 2$.


Uniform universal covers of uniform spaces
Abstract We develop a generalized covering space theory for a class of uniform spaces called coverable spaces. Coverable spaces include all geodesic metric spaces, connected and locally pathwise
Uniform Spaces, I
Publisher Summary Uniform spaces can be defined in various equivalent ways. Every uniform space carries a natural topology; it is defined using neighborhood bases. A uniformly continuous map is also
Covering maps for locally path-connected spaces
We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of
Topological Groups and Related Structures
[i]Topological Groups and Related Structures[/i] provides an extensive overview of techniques and results in the topological theory of topological groups. This overview goes sufficiently deep and is
The fundamental group as a topological group
Abstract This paper is devoted to the study of a natural group topology on the fundamental group which remembers local properties of spaces forgotten by covering space theory and weak homotopy type.
The fundamental group of a compact metric space
We give a forcing free proof of a conjecture of Mycielski that the fundamental group of a connected locally connected compact metric space is either finitely generated or has the power of the
Group Actions and Covering Maps in the Uniform Category
In Rips Complexes and Covers in the Uniform Category (arXiv:0706.3937) we define, following James, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. In
Small loop spaces
Abstract The importance of small loops in the covering space theory was pointed out by Brodskiy, Dydak, Labuz, and Mitra in [2] and [3] . A small loop is a loop which is homotopic to a loop contained
Quotients of uniform spaces
Abstract This paper develops the basic theory of quotients of uniform spaces via sufficiently nice group actions. We generalize and unify two fundamental constructions: quotients of topological
Introduction To Uniform Spaces
Introduction 1. Uniform structures 2. Induced and coinduced uniform structures 3. The uniform topology 4. Completeness and completion 5. Topological groups 6. Uniform transformation groups 7. Uniform