Fundamental Group of Rough Topological Spaces

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.


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].
105 and the lower approximation. Intuitively, the boundary region of the set consists of all elements that cannot be classified uniquely to the set or its complement, by employing available knowledge. Definition 2.5 [23]. A set is said to be a rough set, if it has a non-empty boundary region. If the boundary region is empty, then the set is a crisp or exact set.
Definition 2.10 [3]. Let = ( , ) be a rough subset of the approximation space (1, ). Let 2 and 2 be any two topologies which contain only exact subsets of and respectively. Then the pair 2 = (2 ,2 ) is said to be a rough topology on the rough set = ( , ), the pair ( , 2) is known as a rough topological space (RTS). Also in a rough topology 2 = (2 ,2 ), 2 is known as the lower rough topology and 2 is known as the upper rough topology on . [3]. Let . = (. , . ) be any rough subset of a RTS ( , 2), where = ( , ) and 2 = (2 ,2 Theorem 2.12 [3]. Consider a RTS ( , 2), where = ( , ) and 2 = (2 ,2 ). Let 2 be the collection of all rough open subsets of ( , 2). Then 2 is a topology on .  and 2 = (2 ,2 ) is said to be lower rough closed (LRC) if / 4 5 = ∖ / ∈ 2 . Also / is said to be upper rough closed (URC) if / 4 7 = ∖ / ∈ 2 . / is said to be rough closed (RC) if it is lower rough closed (LRC) and upper rough closed (URC). That is a subset / = (/ , / ) of the RTS ( , 2) is rough closed subset iff its lower approximation is closed with respect to the lower topology and its upper approximation is closed with respect to the upper topology of ( , 2).  (ii) If < is a RP in ;, then the pre-image 8 =# (< ) of 8 defined by

Main Results
is a RP in .
Proof. The proof follows directly from Definition 3.3.  Proof. Let A be rough closed set in (;, σ), since the complement of the pre-image of . is the pre-image of the complement of .. Then Therefore the result follows. Definition 3.7. Let ( , @) be an ordinary topological space. The collection 2̃= B2 C D2 C is rough subset of and 2 S ∈ @T is rough topology (RT) on induced by @. The pair ( , 2) is called the rough topological space induced by ( , @).
Note. If U is the unit interval and V W is an Euclidean subspace topology on U, then (U, V X W ) denotes the RTS induced by the Euclidean space (U, V W ).
Proof. Clearly, ℎ is unique well-defined function. We only need to show that ℎ is rough continuous.
Let \ be a RCS in (;, σ). Definition 3.11. Let = ( , ) and ; = (; , ; ) be two rough sets such that is a rough subset of ;. If d C 57 is a rough characteristic function of , then the collection 2 C = B> C ∩ d C 57 D> C ∈ 2T is rough topology (RT) on , called a rough subspace topology and the pair ( , 2 C ) is called a rough topological subspace (RTi j ) of (;, 2).  Remark. If the initial and terminal points are equal, then we called the RPt ℎ, a rough loop (RL). We denote the collection of all RLs in ( , τ) by o (( , τ), ). The RP is called a rough base point of ( , τ) and (( , τ), ) is called pointed RTS. A continuity of a map between these pointed topological spaces is guaranteed by Definition 3.3. Definition 3.14. Let ( , τ) be RTS and , < be any two rough points in . A RTS ( , τ) is called rough path-connected if there exists an RPt ℎ ∈ ( , τ) with initial and terminal points and < respectively. Definition 3.15. Let ℎ be a rough path (RPt) in ( , 2) having the initial and finial points as and < , respectively. The inverse of ℎ is the rough path ℎ =# defined by ℎ =# (\ ) = ℎ(1 − \ ) for all \ ∈ (U, V X W ), where < and are the initial and finial points, respectively. Then the function k is called a rough homotopy (RH) between ℎ and s. Proof. To show this, it is suffices to show that the following are satisfied: defined by k(u , \ ) = for all u , \ ∈ U is RH between ℎ and ℎ i.e., is a constant function which is continuous, k(0, \ ) = k(1, \ ) = < for all \ ∈ (U, r X W ) and is a constant function.
Then we define the homotopy between (ℎ * s) * x and ℎ * (s * x) as Thus, the result follows.