Algebraic structures on topological spaces can’t distinguish all the non-homeomorphic spaces. Recently, by equipping some of these algebraic structures with topology, one can separate the non-homeomorphic spaces with the same algebraic structures. Particularly, the fundamental group is equipped by the various types of topologies which some of them make it a topological group and some of them don't. Virk and Zastrow has studied the generalizations of the existing topologies on the fundamentalgroup to the universal Path space with a proper comparison. If we remove the condition α(0) = x0 from the definition of the universal path space, we obtain an object that have been discussed in mathematical literatures under the name “ fundamental groupoid”. Indeed, fundamental groupoid denoted by πX , is the category of homotopy classes of paths in X as the morphisms and has the set X as the objects set. For any x , y ∈X , the set πXx,y is the set of homotopy classes of paths in X from x to y . We can consider the object group at x , πXx , as the well-known fundamental group π1X,x.
In order to answer the question how these topologies can be generalized on the fundamental groupoids, the authors has introduced the Lasso topology on the fundamental groupoid of a locally path connected space in which makes it a topological groupoid. A topological groupoid is a groupoid G together with topologies on G and G0 such that the structure maps are continuous.
R. Brown and G. Danesh-Naruie were the first and only ones to take this step. They have defined a topology on a quotient of the fundamental groupoid such that it became a topological groupoid when the given space X is locally path connected and semilocally simply connected.
Here, we introduce whisker topology on the fundamental groupoid of a locally path connected space X in which it’s basis is known and by some assumptions, we can consider it as a generalization of the whisker topology on the fundamental group.
Material and methods
For a given topological space (X,τ) , let [α]∈πX(x,y) , where x,y∈X . If V , W are open neighborhoods of x , y, respectively, one can define
Nα,V,W:=β∈πX β≃γ*α*λ, γI⊆V, λI⊆W} ,
where γ1=α0=x and α1=λ0=y.
Theorem: The family
Nα,V,W;α∈πXx,y,x∈V∈τ,y∈W∈τ
forms a basis for a topology on fundamental groupoid.
The topology that is generated by this basis, is called Whisker topology.
Conclusion
Based on the results that we presented in this paper:
If X is a small loop transfer space;
The multiplication map m: πwhX × πwhX → πwhX is continuous.
The fundamental groupoid with the Whisker topology is a topological groupoid.
The inherited topology from fundamental groupoid πwhX on the object group πwhX(x) equals to the Whisker topology on π1X,x.