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Maximal topologies on groups. (English. Russian original) Zbl 0935.22002
Sib. Math. J. 39, No. 6, 1184-1194 (1998); translation from Sib. Mat. Zh. 39, No. 6, 1368-1381 (1998).
The article is devoted to the notion of a maximal topological group. A topological space without isolated points is called a maximal space if the space has an isolated point in every stronger topology. A topological group is called maximal if the underlying topological space of the group is a maximal space.
The problem of the existence of a maximal topological group is a problem of set theory by nature. However, maximal homogeneous spaces and left-topological groups can be easily constructed in $$ZFC$$.
The author proves that, in $$ZFC$$, an arbitrary infinite group can be endowed with a maximal regular left-invariant topology. In addition, the construction of this topology yields a solution to the Hindman-Strauss problem of a regular idempotent [Semigroup Forum 51, No. 3, 299-318 (1995; Zbl 0843.22005)]. Besides, the author proves some other theorems about groups with strongest left-invariant topology.

##### MSC:
 22A05 Structure of general topological groups 54H11 Topological groups (topological aspects)
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##### References:
 [1] V. I. Malykhin, ”Extremally disconnected and nearly extremally disconnected groups,” Dokl. Akad. Nauk,220, No. 1, 27–30 (1975). [2] I. V. Protasov, ”Filters and topologies on semigroups,” Mat. Studii. Pratsi L’vivsk. Mat. Tovarishch., No. 3, 15–28 (1994). · Zbl 0927.22009 [3] V. I. Malykhin, ”Extremally disconnected topological groups,” Uspekhi Mat. Nauk,34, No. 6, 59–66 (1979). · Zbl 0426.22002 [4] N. Hindman and D. Strauss, ”Nearly prime subsemigroup of {$$\beta$$}$$\mathbb{N}$$, Semigroup Forum,51, No. 3, 299–318 (1995). · Zbl 0843.22005 [5] N. Hindman, ”Ultrafilters and combinatorial number theory,” Lecture Notes in Math.,751, 49–184 (1979). · Zbl 0416.10042 [6] N. Hindman, ”The semigroup {$$\beta$$}$$\mathbb{N}$$ and its applications to number theory,” de Gruyter Exp. Math.,1, 347–360 (1990). · Zbl 0714.11006 [7] N. Hindman, ”Ultrafilters and Ramsey theory–an update,” Lecture Notes in Math.,1401, 97–118 (1989). · Zbl 0701.05060 [8] N. Hindman, ”Recent results on the aligebraic structure of {$$\beta$$}S,” Ann. New York Acad. Sci.,767, 73–84 (1995). · Zbl 0922.54023 [9] R. Ellis, Lectures on Topological Dynamics, Benjamin, New York (1969). · Zbl 0193.51502 [10] D. Davenport and N. Hindman, ”A proof of van Douwen’s right ideal theorem,” Proc. Amer. Math. Soc.,113, No. 2, 573–580 (1991). · Zbl 0735.22002 [11] T. Papazyan, ”Extremal topologies on a semigroup,” Topology Appl.,39, No. 3, 229–243 (1991). · Zbl 0760.22003 [12] E. G. Zelenyuk, Finite Groups in {$$\beta$$}$$\mathbb{N}$$ Are Trivial [Preprint/NAN Ukrainy. Inst. Mat.; No. 96.3], Kiev (1996). · Zbl 0927.22004 [13] W. Ruppert, ”Compact semitopological semigroups: An intrinsic theory,” Lecture Notes in Math.,1079, 1–260 (1984). · Zbl 0606.22001 [14] I. V. Protasov, M. G. Traĉenko, V. V. Trachuk, R. G. Wilson, and I. V. Yaschenko, Almost All Submaximal Groups Are {$$\sigma$$}-Discrete [Preprint/UAM], Mexico (1996). [15] I. V. Protasov, ”Indecomposable topologies on groups,” Ukrain. Mat. Zh.,50, No. 10 (1998) (to appear). · Zbl 0934.22007 [16] I. V. Protasov, ”Decomposability of {$$\tau$$}-bounded groups,” Mat. Studii. Pratsi L’vivsk. Mat. Tovarishch., No. 5, 17–20 (1995). · Zbl 0927.22001 [17] Kourovka Notebook: Unsolved Problems of Group Theory [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk (1995). [18] I. V. Protasov, ”Ultrafilters and topologies on groups,” Sibirsk. Mat. Zh.,34, No. 5, 163–180 (1993). · Zbl 0828.22002 [19] W. W. Comfort, ”Ultrafilters: some old and some new results,” Bull. Amer. Math. Soc.,83, No. 4, 417–455 (1977). · Zbl 0355.54005 [20] V. I. Malykhin and I. V. Protasov, ”Maximal resolvability of bounded groups,” topology Appl.,73, No. 3, 227–232 (1996). · Zbl 0866.54032 [21] S. Shelach, ”Proper forcing,” Lectures Notes in Math., Springer-Verlag, Berlin,940 (1982). [22] A. Blass and N. Hindman, ”On strongly summable ultrafilters and union ultrafilters,” Trans. Amer. Math. Soc.,304, No. 1, 83–97 (1987). · Zbl 0643.03032 [23] E. G. Zelenyuk, ”Topological groups with finite ultrafilter semigroups,” Mat. Studii. Pratsi L’vivsk. Mat. Tovarishch., No. 6, 41–52 (1996). · Zbl 0927.22004 [24] E. G. Zelenyuk and I. V. Protasov, ”Topologies on abelian groups,” Izv. Akad. Nauk SSSR Ser. Mat.,54, No. 5, 1090–1107 (1990). · Zbl 0704.22003
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