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Spanier spaces and covering theory of non-homotopically path Hausdorff spaces
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Thick Spanier groups and the first shape group
- Rocky Mountain Journal of Mathematics
"... We develop a new route through which to explore ker ΨX, the kernel of the pi1−shape group homomorphism determined by a general space X, and establish, for each locally path connected, paracompact Hausdorff space X, ker ΨX is precisely the Spanier group of X. 1 ..."
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We develop a new route through which to explore ker ΨX, the kernel of the pi1−shape group homomorphism determined by a general space X, and establish, for each locally path connected, paracompact Hausdorff space X, ker ΨX is precisely the Spanier group of X. 1
Topological Fundamental Groups and Small Generated Coverings
"... This paper is devoted to study some topological properties of the SG subgroup, pisg1 (X, x), of the quasitopological fundamental group of a based space (X, x), pi qtop 1 (X, x), its topological properties as a subgroup of the topological fundamental group piτ1 (X, x) and its influence on the existen ..."
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This paper is devoted to study some topological properties of the SG subgroup, pisg1 (X, x), of the quasitopological fundamental group of a based space (X, x), pi qtop 1 (X, x), its topological properties as a subgroup of the topological fundamental group piτ1 (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.
Open subgroups of free topological groups
, 2014
"... The theory of covering spaces is often used to prove the Nielsen-Schreier theorem, which states that every subgroup of a free group is free. We ap-ply the more general theory of semicovering spaces to obtain analogous subgroup theorems for topological groups: Every open subgroup of a free Graev topo ..."
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The theory of covering spaces is often used to prove the Nielsen-Schreier theorem, which states that every subgroup of a free group is free. We ap-ply the more general theory of semicovering spaces to obtain analogous subgroup theorems for topological groups: Every open subgroup of a free Graev topological group is a free Graev topological group. An open sub-group of a free Markov topological group is a free Markov topological group if and only if it is disconnected. 1
On the Spanier Groups and Covering and Semicovering Spaces
"... For a connected, locally path connected space X, let H be a subgroup of the funda-mental group of X, pi1(X, x). We show that there exists an open cover U of X such that H contains the Spanier group pi(U, x) if and only if the core of H in pi1(X, x) is open in the quasitopological fundamental group p ..."
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For a connected, locally path connected space X, let H be a subgroup of the funda-mental group of X, pi1(X, x). We show that there exists an open cover U of X such that H contains the Spanier group pi(U, x) if and only if the core of H in pi1(X, x) is open in the quasitopological fundamental group piqtop1 (X, x) or equivalently it is open in the topological fundamental group piτ1 (X, x). As a consequence, using the relation between the Spanier groups and covering spaces, we give a classification for connected covering spaces of X based on the conjugacy classes of subgroups with open core in piqtop1 (X, x). Finally, we give a necessary and sufficient condition for the existence of a semicovering. Moreover, we present a condition under which every semicovering of X is a covering.
