Fundamental Group of Rough Topological Spaces

Laaro Abdullateef

doi:10.34198/ejms.5121.103119

Abstract

In this paper we study and define the concept of the fundamental group of rough topological spaces (RTSs), which deeply depends on the concepts of rough sets (RSs) and rough topology (RT). Working towards this stated objective, we define the concept of rough path (RPt) which gives room for the introduction of rough loop (RL). We also define the concepts of rough homotopy (RH) and later shows that it is indeed an equivalence relation. We introduce the fundamental group of rough topological spaces by showing that all the group axioms satisfied. Also, this paper establish the fact that most of the results in fundamental group of ordinary topological spaces are also hold for the fundamental group of rough topological spaces.

Highlights

IntroductionDue to the applications of classical method in solving various types of inexact or uncertainties problems in economics, engineering and environment, several theories which include: the theory of probability, the theory of fuzzy sets, theory of rough sets and the interval mathematics are introduced as mathematical tools for dealing with these uncertainties [8]
Due to the applications of classical method in solving various types of inexact or uncertainties problems in economics, engineering and environment, several theories which include: the theory of probability, the theory of fuzzy sets, theory of rough sets and the interval mathematics are introduced as mathematical tools for dealing with these uncertainties [8].Rough sets was proposed by Pawlak [21] as a useful tool to deal with uncertainty and incomplete information and it has been approved to be effective approach to intelligent systems characterized by insufficient and incomplete information [26]
Considering the fact that in recent years many researchers have considered fundamental group introduced by Poincare [2] in the study of other mathematical approach, which includes the fundamental group as a topological group [9], the fundamental of intuitionistic fuzzy topological spaces and its algebraic properties [7, 15], the fundamental group of quotient spaces [13], the Fuzzy ∗-Fundamental Group of Fuzzy ∗-Structure Spaces [18], monoids in the fundamental groups of the complement of logarithmic free divisors in [14]

Summary

Introduction

Due to the applications of classical method in solving various types of inexact or uncertainties problems in economics, engineering and environment, several theories which include: the theory of probability, the theory of fuzzy sets, theory of rough sets and the interval mathematics are introduced as mathematical tools for dealing with these uncertainties [8]. Since rough sets and its applications have attracted the interest of researchers in many fields [10] This includes, the generalized multi-fuzzy rough sets and the induced topology [9], rough set theory for topological spaces [8], generalized rough sets based on neighborhood systems. Considering the fact that in recent years many researchers have considered fundamental group introduced by Poincare [2] in the study of other mathematical approach, which includes the fundamental group as a topological group [9], the fundamental of intuitionistic fuzzy topological spaces and its algebraic properties [7, 15], the fundamental group of quotient spaces [13], the Fuzzy ∗-Fundamental Group of Fuzzy ∗-Structure Spaces [18], monoids in the fundamental groups of the complement of logarithmic free divisors in [14]. The applications of fundamental group to the study of an effective approach to intelligent systems, i.e., rough topological spaces will be our priority in this paper

Preliminaries

Main Results

Conclusion