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A completion of \(\mathbb Z\) is a field. (English) Zbl 1080.54500

Summary: We define various ring sequential convergences on  \(\mathbb Z\) and \(\mathbb Q\). We describe their properties and properties of their convergence completions. In particular, we define a convergence \(\mathbb L_1\) on  \(\mathbb Z\) by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields  \(\mathbb Z/(p)\). Further, we show that \((\mathbb Z, \mathbb L^\ast _1)\) is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D.Dikranjan.

MSC:

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
13J99 Topological rings and modules
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References:

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