Sequentially complete groups: Dimension and minimality. (English) Zbl 0976.22002
A topological group \(G\) is minimal if every continuous isomorphism \(G\to H\) to a Hausdorff group \(H\) is open; it is called sequentially complete iff every Cauchy sequence in \(G\) converges. The authors discuss sequentially complete topological groups in the perspective of dimension/connectedness and minimality.
The main result states: A sequentially complete minimal Abelian group \(G\) contains the connected component of its completion whenever the connected component of \(G\) is of nonmeasurable size. – It has also been proved that the hereditarily disconnected sequentially complete minimal Abelian groups split in direct product of primary components and the sequentially complete totally minimal Abelian groups are compact.
The main result states: A sequentially complete minimal Abelian group \(G\) contains the connected component of its completion whenever the connected component of \(G\) is of nonmeasurable size. – It has also been proved that the hereditarily disconnected sequentially complete minimal Abelian groups split in direct product of primary components and the sequentially complete totally minimal Abelian groups are compact.
Reviewer: S.Ganguly (Kolkata)
MSC:
| 22A05 | Structure of general topological groups |
| 54D25 | “\(P\)-minimal” and “\(P\)-closed” spaces |
| 22B05 | General properties and structure of LCA groups |
| 54H11 | Topological groups (topological aspects) |
| 54A35 | Consistency and independence results in general topology |
| 54D30 | Compactness |
| 54H13 | Topological fields, rings, etc. (topological aspects) |
Keywords:
topological group; Hausdorff group; sequentially complete topological groups; hereditarily disconnected sequentially complete minimal Abelian groups; sequentially complete totally minimal Abelian groups are compactReferences:
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