## 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)

Zbl 0309.22004
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### References:

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