Topological fundamental groups and small generated coverings

Abstract

This paper is devoted to study some topological properties of the SG subgroup, ${\pi }_1^{sg}(X,x)$, of the quasitopological fundamental group of a based space $(X,x)$, $\pt$, its topological properties as a subgroup of the topological fundamental group ${\pi }_1^{\tau }(X,x)$ and its influence on the existence of universal covering of $X$. First, we introduce small generated spaces which have indiscrete topological fundamental groups and also small generated coverings which are universal coverings in the categorical sense. Second, we give a necessary and sufficient condition for the existence of the small generated coverings. Finally, by introducing the notion of semi-locally small generatedness we show that the quasitopological fundamental groups of semi-locally small generated spaces are topological groups.

13 pages, 1 figure, to appear in Mathematica Slovaca

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• On semilocally simply connected spaces
• The topological fundamental group and free topological groups
• Discreteness and Homogeneity of the Topological Fundamental Group
• On Topological Homotopy Groups of $n$-Hawaiian like spaces
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Cited by in corpus (8)

• On the Spanier Groups and Covering and Semicovering Spaces
• On topological fundamental groups of quotient spaces
• On Subgroup Topologies on Fundamental Groups
• Comparison of Topologies on Fundamental Groups with Subgroup Topology Viewpoint
• On Strong Small Loop Transfer Spaces Relative to Subgroups of Fundamental Groups
• Small Loop Transfer Spaces with Respect to Subgroups of Fundamental Groups
• On Subgroups of Topologized Fundamental Groups and Generalized Coverings
• On quasi-small loop groups