Topologizations of \(G\)-spaces. (English. Russian original) Zbl 1347.54052

Banakh, Taras; Protasov, Igor

J. Math. Sci., New York 188, No. 2, 77-84 (2013); translation from Ukr. Mat. Visn. 9, No. 3, 308-317 (2012).

Summary: For a \(G\)-space \(X\), we put and explore a question whether \(X\) admits a non-discrete Hausdorff \(G\)-invariant topology.

MSC:

54H11	Topological groups (topological aspects)
22F05	General theory of group and pseudogroup actions

Keywords:

\(G\)-space; \(G\)-invariant topology; filter

References:

[1]	V. I. Arnautov, ”Nondiscrete topologizability of countable rings,” Dokl. Akad. Nauk SSSR, 191, 747–750 (1970). · Zbl 0209.33903
[2]	T. Banakh, I. Protasov, and O. Sipacheva, Topologizations of Sets Endowed with an Action of Monoid, preprint available at ArXiv:1112.5729v1. · Zbl 1294.54020
[3]	D. Dikranjan and I. Protasov, ”Counting maximal topologies on countable groups and rings,” Topology Appl., 156, 322–325 (2008). · Zbl 1171.54029 · doi:10.1016/j.topol.2008.07.009
[4]	D. Dikranjan and D. Shakhmatov, ”The Markov–Zariski topology of an Abelian group,” J. Algebra, 324, No. 6, 1125–1158 (2010). · Zbl 1211.22002 · doi:10.1016/j.jalgebra.2010.04.025
[5]	N. Hindman, I. Protasov, and D. Strauss, ”Topologies on S determined by idempotents in {\(\beta\)}S,” Topol. Proceed., 23, 155–190 (1998). · Zbl 0970.54036
[6]	A. A. Klyachko and A. V. Trofimov, ”The number of non-solutions of an equation in a group,” J. Group Theory, 8, No. 6, 747–754 (2005). · Zbl 1101.20021
[7]	A. A. Markov, ”On unconditionally closed sets,” Mat. Sb., 18, No. 1, 3–28 (1946). · Zbl 0061.04208
[8]	J. van Mill, ”A note on Ford example,” Topol. Proceed., 28, 689–694 (2004). · Zbl 1088.54015
[9]	S. Morris and V. Obraztsov, ”Nondiscrete topological groups with many discrete subgroups,” Topol. Appl., 84, 103–120 (1998). · Zbl 0933.22003
[10]	A. Yu. Ol’shanskii, ”A remark on a countable nontopologized group,” Vest. Mosk. Univ. Ser. I. Mat. Mekh., No. 3, 103 (1980).
[11]	A. Yu. Ol’shanskii, The Geometry of Defining Relations in Groups [in Russian], Nauka, Moscow, 1989.
[12]	I. V. Protasov, ”Filters and topologies on semigroups,” Mat. Stud., 3, 15–28 (1994). · Zbl 0927.22009
[13]	I. V. Protasov, ”Extremal topologies on groups,” Mat. Stud., 15, 9–22 (2001). · Zbl 0989.22003
[14]	I. V. Protasov, ”Maximal topologies on groups,” Sib. Math. J., 39, 1368–1381 (1998). · Zbl 0935.22002
[15]	I. Protasov and E. Zelenyuk, Topologies on Groups Determined by Sequences, VNTL, Lviv, 1999. · Zbl 0977.54029
[16]	S. Shelah, ”On a problem of Kurosh, Jo’nsson groups and applications,” Word Problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), North-Holland, Amsterdam, 1980, pp. 373–394.
[17]	A. V. Trofimov, ”A theorem on embedding in nontopologizable groups,” Vest. Mosk. Univ. Ser. I. Mat. Mekh., No. 3, 60–62 (2005). · Zbl 1106.20020
[18]	A. V. Trofimov, ”A perfect nontopologizable group,” Vest. Mosk. Univ. Ser. I. Mat. Mekh., No. 1, 7–13 (2007). · Zbl 1164.20011
[19]	Y. Zelenyuk, ”On topologizing groups,” J. Group Theory, 10, No. 2, 235–244 (2007). · Zbl 1115.22001