Relatively coarse sequential convergence. (English) Zbl 0897.54002

Frič, R.; Zanolin, F.

Czech. Math. J. 47, No. 3, 395-408 (1997).

Summary: We generalize the notion of a coarse sequential convergence compatible with an algebraic structure to a coarse one in a given class of convergences. In particular, we investigate coarseness in the class of all compatible convergences (with unique limits) the restriction of which to a given subset is fixed. We characterize such convergences and study relative coarseness in connection with extensions and completions of groups and rings. E.g., we show that: (i) each relatively coarse dense group precompletion of the group of rational numbers (equipped with the usual metric convergence) is complete; (ii) there are exactly exp exp \(\omega \) such completions; (iii) the real line is the only one of them the convergence of which is Fréchet. Analogous results hold for the relatively coarse dense field precompletions of the subfield of all complex numbers both coordinates of which are rational numbers.

MSC:

54A20	Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54H11	Topological groups (topological aspects)
54H13	Topological fields, rings, etc. (topological aspects)
16W99	Associative rings and algebras with additional structure
20K35	Extensions of abelian groups
20K45	Topological methods for abelian groups

Keywords:

compatible relatively coarse sequential convergence; FLUSH-group; FLUSH-ring; completion; extension

References:

[1]	Banaschewski, B.: Minimal topological algebras. Math. Ann. 211 (1974), 107-114. · Zbl 0275.46043 · doi:10.1007/BF01344165
[2]	Contessa, M. and F. Zanolin: On some remarks about a not completable convergence ring. General Topology and its Relations to Modern Analysis and Algebra V. Proc. Fifth Prague Topological Sympos., Prague 1981), Heldermann Verlag, Berlin, 1982, pp. 107-114.
[3]	Dikranjan, D., Frič, R. and F. Zanolin: On convergence groups with dense coarse subgroups. Czechoslovak Math. J. 37 (1987), 471-479. · Zbl 0637.22002
[4]	Frič, R.: Products of coarse convergence groups. Czechoslovak Math. J. 38 (1988), 285-290. · Zbl 0663.54001
[5]	Frič, R.: Rationals with exotic convergences. Math. Slovaca 39 (1989), 141-147. · Zbl 0678.54001
[6]	Frič, R.: Rationals with exotic convergences II. Math. Slovaca 40 (1990), 389-400. · Zbl 0776.54029
[7]	Frič, R.: On ring convergences. Riv. Mat. Pura Appl. 11 (1992), 125-138. · Zbl 0774.16016
[8]	Frič, R. and V. Koutník: Completions for subcategories of convergence rings. Categorical Topology and its Relations to Modern Analysis, Algebra and Combinatories, World Scientific Publishing Co., Singapore, 1989, pp. 195-207.
[9]	Frič, R. and F. Zanolin: Coarse convergence groups. Convergence Structures 1984 (Proc. Conf. on Convergence, Bechyně 1984), Akademie-Verlag Berlin, 1985, pp. 107-114.
[10]	Frič, R. and F. Zanolin: Sequential convergence in free groups. Rend. Istit. Mat. Univ. Trieste 18 (1986), 200-218. · Zbl 0652.22001
[11]	Frič, R. and F. Zanolin: Coarse sequential convergence in groups, etc. Czechoslovak Math. J. 40 (1990), 459-467. · Zbl 0747.54002
[12]	Frič, R. and F. Zanolin: Strict completions of \(L_0^*\)-groups. Czechoslovak Math. J. 42 (1992), 589-598. · Zbl 0797.54007
[13]	Frič, R. and F. Zanolin: Minimal pseudotopological groups. (Presented at Categorical Topology and its Relation to Analysis, Algebra and Combinatorics, Praha 1988, unpublished.). (19\(\infty \)).
[14]	Jakubík, J.: On convergence in linear spaces. Mat.-Fyz. Časopis Slovensk. Akad. Vied 6 (1956), 57-67.
[15]	Novák, J.: On completion of convergence commutative groups. General Topology and its Relations to Modern Analysis and Algebra III (Proc. Third Prague Topological Sympos., 1971), Academia, Praha, 1972, pp. 335-340.
[16]	Simon, P. and F. Zanolin: A coarse convergence group need not be precompact. Czechoslovak Math. J. 37 (1987), 480-486. · Zbl 0637.22003