Ursul, M. I. Imbedding a topological group as a quasi-component into another group. (Russian) Zbl 0652.22002 Mat. Issled. 91, 92-102 (1987). The quasicomponent \(Q_ x\) of a point x in a topological space X is the intersection of all its clopen neighbourhoods. Let \(\tau\) (\(\geq 1)\) be a transfinite number. The \(\tau\)-quasicomponent \(Q_{x,\tau}(X)\) of x in X is then defined inductively by \(Q_{x,1}(X)=Q_ x\), and \(Q_{x,\tau +1}(X)\) is \(Q_{x,1}(Q_{x,\tau}(X))\) for successor ordinals, the intersection of all \(Q_{x,\lambda}(X)\) for \(\lambda <\tau\), otherwise. It is shown that any Abelian topological group can be embedded as the \(\tau\)-quasicomponent of the identity in another Abelian topological group, but that this result does not extend to the non-Abelian case. Moreover, if \(1\leq \lambda <\tau\) and S is any Abelian topological group, there is an Abelian topological group G such that \(Q_{0,\tau}(G)=S\neq Q_{0,\lambda}(G)\). Reviewer: D.L.Grant MSC: 22A05 Structure of general topological groups 54B05 Subspaces in general topology Keywords:quasicomponent; Abelian topological group; \(\tau\)-quasicomponent of the identity PDFBibTeX XMLCite \textit{M. I. Ursul}, Mat. Issled. 91, 92--102 (1987; Zbl 0652.22002) Full Text: EuDML