Properties of Algebraic Spaces
Properties of Algebraic Spaces

... In other words, the small Zariski site XZar of X is canonically identified with a site associated to the topological space |X| (see Sites, Example 6.4). Proof. In order to prove (1) let us construct the inverse of the rule. Namely, suppose that W ⊂ |X| is open. Choose a presentation X = U/R correspo ...
1 Introduction
1 Introduction

Fuzzy Strongly Locally Connected Space By Hanan Ali
Fuzzy Strongly Locally Connected Space By Hanan Ali

ro-PDF - University of Essex
ro-PDF - University of Essex

Totally supra b−continuous and slightly supra b−continuous functions
Totally supra b−continuous and slightly supra b−continuous functions

Maximal compact subgroups in the o-minimal setting
Maximal compact subgroups in the o-minimal setting

... (i) G has maximal definable definably compact subgroups. (ii) G = K · N (G), for some definable definably compact subgroup K < G. (iii) A(G) = {e}. Proof. (i) ⇒ (iii): Take K to be a maximal definable definably compact subgroup, and T a 0-Sylow of G. Suppose, for a contradiction, that A(G) is infini ...
CHARACTERIZATIONS OF FUZZY α
CHARACTERIZATIONS OF FUZZY α

Local entropy theory - School of Mathematical Sciences
Local entropy theory - School of Mathematical Sciences

Proper Morphisms, Completions, and the Grothendieck Existence
Proper Morphisms, Completions, and the Grothendieck Existence

Properties of Algebraic Stacks
Properties of Algebraic Stacks

On supra λ-open set in bitopological space
On supra λ-open set in bitopological space

Chapter 1 Sheaf theory
Chapter 1 Sheaf theory

FUZZY ORDERED SETS AND DUALITY FOR FINITE FUZZY
FUZZY ORDERED SETS AND DUALITY FOR FINITE FUZZY

Peterzil
Peterzil

Andr´e-Quillen (co)Homology, Abelianization and Stabilization
Andr´e-Quillen (co)Homology, Abelianization and Stabilization

Coarse Structures on Infinite Groups
Coarse Structures on Infinite Groups

THE SMOOTH IDEAL Introduction A measurable space is a set
THE SMOOTH IDEAL Introduction A measurable space is a set

I. Topological background
I. Topological background

S-CLUSTER SETS IN FUZZY TOPOLOGICAL SPACES 1. Introduction
S-CLUSTER SETS IN FUZZY TOPOLOGICAL SPACES 1. Introduction

SYMMETRIC SPECTRA Contents Introduction 2 1
SYMMETRIC SPECTRA Contents Introduction 2 1

I-fuzzy Alexandrov topologies and specialization orders
I-fuzzy Alexandrov topologies and specialization orders

QUOTIENT SPACE OF LMC
QUOTIENT SPACE OF LMC

On S-closed and Extremally Disconnected Fuzzy Topological Spaces
On S-closed and Extremally Disconnected Fuzzy Topological Spaces

Supra b-compact and supra b
Supra b-compact and supra b

Properties of Fuzzy Total Continuity ∗
Properties of Fuzzy Total Continuity ∗

Fundamental group

In the mathematics of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a topological invariant: homeomorphic topological spaces have the same fundamental group.Fundamental groups can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the associated universal covering space. The abelianization of the fundamental group can be identified with the first homology group of the space. When the topological space is homeomorphic to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations.Henri Poincaré defined the fundamental group in 1895 in his paper ""Analysis situs"". The concept emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Poincaré, and Felix Klein. It describes the monodromy properties of complex-valued functions, as well as providing a complete topological classification of closed surfaces.