×

Imbedding a topological group as a quasi-component into another group. (Russian) Zbl 0652.22002

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
PDFBibTeX XMLCite
Full Text: EuDML