Dense subsets of maximally almost periodic groups

Comfort, W.; García-Ferreira, Salvador

doi:10.1090/S0002-9939-00-05557-X

Dense subsets of maximally almost periodic groups
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by W. W. Comfort and Salvador García-Ferreira PDF

Proc. Amer. Math. Soc. 129 (2001), 593-599 Request permission

Abstract:

A (discrete) group $G$ is said to be maximally almost periodic if the points of $G$ are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology $\mathcal {T}$ on a group $G$ is totally bounded if whenever $\emptyset \neq U\in \mathcal {T}$ there is $F\in [G]^{<\omega }$ such that $G=UF$. For purposes of this abstract, a family $\mathcal {D}\subseteq \mathcal {P}(G)$ with $(G,\mathcal {T})$ a totally bounded topological group is a strongly extraresolvable family if (a) $|\mathcal {D}|>|G|$, (b) each $D\in \mathcal {D}$ is dense in $G$, and (c) distinct $D,E\in \mathcal {D}$ satisfy $|D\cap E|<d(G)$; a totally bounded topological group with such a family is a strongly extraresolvable topological group. We give two theorems, the second generalizing the first. Theorem 1. Every infinite totally bounded group contains a dense strongly extraresolvable subgroup. Corollary. In its largest totally bounded group topology, every infinite Abelian group is strongly extraresolvable. Theorem 2. Let $G$ be maximally almost periodic. Then there are a subgroup $H$ of $G$ and a family $\mathcal {D}\subseteq \mathcal {P}(H)$ such that (i) $H$ is dense in every totally bounded group topology on $G$; (ii) the family $\mathcal {D}$ is a strongly extraresolvable family for every totally bounded group topology $\mathcal {T}$ on $H$ such that $d(H,\mathcal {T})=|H|$; and (iii) $H$ admits a totally bounded group topology $\mathcal {T}$ as in (ii). Remark. In certain cases, for example when $G$ is Abelian, one must in Theorem 2 choose $H=G$. In certain other cases, for example when the largest totally bounded group topology on $G$ is compact, the choice $H=G$ is impossible.

References

H. Auerbach, Sur les groupes linéaires bornés, Studia Math. 5 (1934), 43–49.
J. G. Ceder, On maximally resolvable spaces, Fund. Math. 55 (1964), 87–93. MR 163279, DOI 10.4064/fm-55-1-87-93
W. W. Comfort and Salvador García-Ferreira, Strongly extraresolvable groups and spaces, Proc. Mexico City Summer (1998) Topology Conference. Topology Proceedings 23. To appear.
W. W. Comfort and Jan van Mill, Groups with only resolvable group topologies, Proc. Amer. Math. Soc. 120 (1994), no. 3, 687–696. MR 1209097, DOI 10.1090/S0002-9939-1994-1209097-X
E. K. van Douwen, The maximal totally bounded group topology on ${G}$ and the biggest minimal ${G}$-space for Abelian groups ${G}$, Topology Appl. 34 (1990), 69–91.
S. García-Ferreira, V. I. Malykhin and A. Tomita, Extraresolvable spaces. Topology and Its Applications. 1999. To Appear.
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2
Albert Eagle, Series for all the roots of a trinomial equation, Amer. Math. Monthly 46 (1939), 422–425. MR 5, DOI 10.2307/2303036
Karl H. Hofmann and Sidney A. Morris, The structure of compact groups, De Gruyter Studies in Mathematics, vol. 25, Walter de Gruyter & Co., Berlin, 1998. A primer for the student—a handbook for the expert. MR 1646190
V. I. Malykhin, Irresolvability is not descriptively good. Manuscript submitted for publication. 1996.
V. I. Malykhin, Something new about totally bounded groups, Mat. Zametki 65 (1999), 474–476. [In Russian].
V. I. Malykhin, To the combinatorial analysis of infinite groups, Manuscript in Preparation. 1998.
V. I. Malykhin and I. V. Protasov, Maximal resolvability of bounded groups, Topology Appl. 73 (1996), no. 3, 227–232. MR 1419796, DOI 10.1016/S0166-8641(96)00020-X
J. von Neumann, Almost periodic functions in a group I, Trans. Amer. Math. Soc. 36 (1934), 445–492.
Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
Dieter Remus, Minimal and precompact group topologies on free groups, Proceedings of the Conference on Locales and Topological Groups (Curaçao, 1989), 1991, pp. 147–157. MR 1100513, DOI 10.1016/0022-4049(91)90014-S
B. L. van der Waerden, Stetigkeitssätze für halbeinfache Liesche Gruppen, Math. Zeitschrift 36 (1933), 780–786.