In this paper, by introducing some kind of small loop transfer spaces at a point, we study the behavior of topologized fundamental groups with the compact-open topology and the whisker topology, π qtop 1 (X, x 0 ) and π wh 1 (X, x 0 ), respectively. In particular, we give necessary or sufficient conditions for coincidence and being topological group of these two topologized fundamental groups. Finally, we give some examples to show that the reverse of some of these implications do not hold, in general.
It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental group and use them to classify coverings, semicoverings, and generalized coverings of a topological space. To do this, we use the concept of subgroup topology on a group and discuss their properties. In particular, we explore which of these topologies make the fundamental group a topological group. Moreover, we provide some examples of topological spaces to compare topologies of fundamental groups.
It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental group and use them to classify coverings, semicoverings, and generalized coverings of a topological space. To do this, we use the concept of subgroup topology on a group and discuss their properties. In particular, we explore which of these topologies make the fundamental group a topological group. Moreover, we provide some examples of topological spaces to compare topologies of fundamental groups.
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