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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

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

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.

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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