# TOPOLOGICAL AND UNIFORM STRUCTURES ON UNIVERSAL COVERING SPACES

@article{Brodskiy2012TOPOLOGICALAU, title={TOPOLOGICAL AND UNIFORM STRUCTURES ON UNIVERSAL COVERING SPACES}, author={N. Brodskiy and Jerzy Dydak and B. Labuz and A. Mitra}, journal={arXiv: Algebraic Topology}, year={2012} }

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…

## 14 Citations

On Topologized Fundamental Groups and Covering Groups of Topological Groups

- Mathematics
- 2020

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

- Mathematics
- 2018

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

- Mathematics
- 2010

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

- Mathematics
- 2018

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

- MathematicsJournal 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

- Mathematics
- 2020

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

- Mathematics, PhysicsActa Mathematica Vietnamica
- 2018

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

- MathematicsTopology and its Applications
- 2019

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

- Mathematics
- 2016

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

- MathematicsHacettepe Journal of Mathematics and Statistics
- 2020

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 $n \ge 2$.…

## References

SHOWING 1-10 OF 27 REFERENCES

Uniform universal covers of uniform spaces

- Mathematics
- 2007

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

- Mathematics
- 2003

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

- Mathematics
- 2008

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

- Mathematics
- 2008

[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

- Mathematics
- 2010

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

- Mathematics
- 1998

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

- Mathematics
- 2008

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

- Mathematics
- 2010

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

- Mathematics
- 2006

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

- Mathematics
- 1987

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…