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Strict completions of \({\mathcal L}^*_ 0\)-groups. (English) Zbl 0797.54007

An \({\mathcal L}^*_ 0\)-group is a group \(G\) with a sequential convergence \(L \subset G^ N \times G\) such that the group operations are sequentially continuous. The notion of a strict \({\mathcal L}^*_ 0\)-group completion of an \({\mathcal L}^*_ 0\)-group is introduced. It is shown that the Novak completion \(\nu G\) [J. Novák, Gen. Topol. Relat. mod. Anal. Algebra III, Proc. 3rd Prague Topol. Symp. 1971, 335-340 (1972; Zbl 0309.22004)] of any abelian \({\mathcal L}^*_ 0\)-group \(G\) is the finest strict \({\mathcal L}^*_ 0\)-group completion of \(G\). A sufficient condition for the existence of the coarsest strict \({\mathcal L}^*_ 0\)-group completion of an abelian \({\mathcal L}^*_ 0\)-group is also given. The main result (which answers a question of J. Novak) states that there are exactly \(2^{2^ \omega}\) nonhomeomorphic strict \({\mathcal L}^*_ 0\)-group completions of \(\mathbb{Q}\). The set \(\mathbb{R}\) of real numbers with the usual metric convergence is the coarsest strict completion of \(\mathbb{Q}\).

MSC:

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54H11 Topological groups (topological aspects)

Citations:

Zbl 0309.22004
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References:

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