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On the topology of topological fundamental groupoids
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This paper is devoted to the Lasso and the Whisker topology of the fundamental groupoid. We prove that topological fundamental groupoid of a given space X is locally path connected in general and is path connected if X is simply connected. We show that for locally path connected space X, the unit map 1 : X → π X is an embedding if and only if X is a semilocally simply connected space. Also, we give conditions that guarantee Hausdorffness of the topological fundamental groupoid.
Title: On the topology of topological fundamental groupoids
Description:
This paper is devoted to the Lasso and the Whisker topology of the fundamental groupoid.
We prove that topological fundamental groupoid of a given space X is locally path connected in general and is path connected if X is simply connected.
We show that for locally path connected space X, the unit map 1 : X → π X is an embedding if and only if X is a semilocally simply connected space.
Also, we give conditions that guarantee Hausdorffness of the topological fundamental groupoid.
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