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Pakdaman A, Shahini F. Fundamental groupoid and Whisker topology. mmr 2023; 9 (1) :55-71
URL: http://mmr.khu.ac.ir/article-1-3132-en.html
URL: http://mmr.khu.ac.ir/article-1-3132-en.html
1- , a.pakdaman@gu.ac.ir
Abstract: (825 Views)
Introduction
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) =x 0 π X X X x y ∈ X π X x , y X x y x π X x π 1 X , 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 groupoidG G G 0
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 spaceX
Here, we introduce whisker topology on the fundamental groupoid of a locally path connected space
Material and methods
For a given topological space( X , τ ) [ α ]∈ π ( x , y ) x , y ∈ X V W x
N α , V , W := β ∈ πX β ≃ γ * α * λ , γ I ⊆ V , λ I ⊆ W }
whereγ 1 = α 0 = x α 1 = λ 0 = y .
Theorem: The family
N α , V , W ; α ∈ πX x , 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:
IfX
The multiplication mapm : π w h X × π w h X → π w h X
The fundamental groupoid with the Whisker topology is a topological groupoid.
The inherited topology from fundamental groupoidπ w h X π w h X ( x ) π 1 X , x .
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) =
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
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
Here, we introduce whisker topology on the fundamental groupoid of a locally path connected space
Material and methods
For a given topological space
where
Theorem: The family
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
The multiplication map
The fundamental groupoid with the Whisker topology is a topological groupoid.
The inherited topology from fundamental groupoid
Type of Study: Research Paper |
Subject:
Mat
Received: 2020/09/27 | Revised: 2024/01/7 | Accepted: 2021/07/11 | Published: 2023/06/20 | ePublished: 2023/06/20
Received: 2020/09/27 | Revised: 2024/01/7 | Accepted: 2021/07/11 | Published: 2023/06/20 | ePublished: 2023/06/20
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