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Completeness of sequential convergence groups. (English) Zbl 0546.54006

A convergence \({\mathfrak L}\) on a set L is a set of pairs \((<x_ n>,x)\), where \(<x_ n>\) is a sequence from L and where \({\mathfrak L}\) satisfies the axioms: (i) if \((<x_ n,x>)\) and \((<x_ n,y>)\) are in \({\mathfrak L}\), then \(x=y\), (ii) \((<x>,x)\) is in \({\mathfrak L}\) for each x in L, and, (iii) if \((<x_ n>,x)\) is in \({\mathfrak L}\) and \(<n_ i>\) is a subsequence of \(<n>\), then \((<x_{n_ i},x>)\) is in \({\mathfrak L}\). A convergence is said to be maximal, and is denoted by \({\mathfrak L}^*\), if it also satisfies (iv) if \(<x_ n>\) is a sequence, if x is in L, and if each subsequence \(<n_ i>\) of \(<n>\) has a subsequence \(<n_{i_ j}>\) such that \((<x_{n_{i_ j}}>,x)\) is in \({\mathfrak L}\), then \((<x_ n>,x)\) is in \({\mathfrak L}\). A convergence has an associated closure operator \(\lambda\). If \((L,+)\) is also a group and if the mapping (x,y)\(\to x-y\) is sequentially continuous, then (L,\({\mathfrak L},\lambda,+)\) is called a convergence group. In a series of four papers, J. Novák studied convergence groups, especially \(C^*\) groups, that is, convergence groups in which the convergence is maximal. He defined Cauchy sequences and constructed a completion for \(C^*\) groups. The author extends the theory to general convergence groups. Three plausible definitions are considered for Cauchy sequences; these are distinct in general, but coincide for \(C^*\) groups. Only one of the definitions forces convergent sequences to be Cauchy sequences. Naturally this is the definition chosen, and the process of completion is under way. Every convergence has a ”minimal completion”: the convergence \({\mathfrak L}\) is replaced by \({\mathfrak L}^*\), the resulting C-group is then a \(C^*\) group, and is completed using Novák’s technique. The minimal completion of a Fréchet convergence group need not be a Fréchet convergence group; a criterion is given for this to occur. The author provides many helpful examples.
Reviewer: F.Carroll

MSC:

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
22A05 Structure of general topological groups
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