×

zbMATH — the first resource for mathematics

Dieudonné completion and \(b_{f}\)-group actions. (English) Zbl 1106.22003
The problems of existence and realization of equivariant compactifications in topological dynamics have been dealt with by many authors. Let \(G\) be a Hausdorff topological group continuously acting on a Tychonoff space \(X\). The main result in this paper (Theorem 3.6) asserts that, if \(G\) is a \(b_f\)-space, and \(X\) is pseudocompact, then there exist such compactifications for \(X\) (i. e. \(X\) can be equivariantly embedded into a compact \(G\)-space as a dense subset; this property is usually referred to as “\(X\) is \(G\)-Tychonoff”), and the topological space underlying the maximal one is the Stone-Čech’s compactification \(\beta X\) of \(X\). (A \(b_f\)-space is characterized by the following property: if restrictions to bounded subsets of a real-valued function defined on such a space are continuous, then the function is continuous.)
The authors deal also with the converse question: what can be said of a space which is \(G\)-Tychonoff and whose maximal equivariant compactification is Stone-Čech’s? They observe that the proof of an important result in this direction, proved by J. de Vries [Topology theory and applications, 5th Colloq., Eger/Hung. 1983, Colloq. Math. Soc. János Bolyai 41, 655–666 (1985; Zbl 0598.54019)], does not use this reference’s initial assumption of \(G\) being locally compact, and thus derive, among other results, that \(X\) is pseudocompact whenever \(G\) is metrizable and the set of elements in \(X\) with open isotropy group has empty interior (Corollary 3.10).
Along the way to these results the authors prove a few interesting facts concerning free topological groups and \(b_f\)-spaces, e. g. both the free topological group and the free Abelian topological group over a pseudocompact space are \(b_f\)-spaces (Theorem 2.4).

MSC:
22A10 Analysis on general topological groups
54H20 Topological dynamics (MSC2010)
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54H15 Transformation groups and semigroups (topological aspects)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Antonyan, N., Equivariant embeddings and compactifications of free G-spaces, Internat. J. math. sci., 1, 1-14, (2003) · Zbl 1014.54024
[2] Antonyan, N.; Antonyan, S.A., Free G-spaces and maximal equivariant compactifications, Ann. mat. pura appl. (4), 184, 3, 407-420, (2005) · Zbl 1097.54030
[3] Antonyan, S.A., Pseudocompact and G-hewitt spaces, Uspekhi mat. nauk, 35, 6 (216), 151-152, (1980) · Zbl 0458.54028
[4] Antonyan, S.A., New proof of the existence of a bicompact G-extension, Comment. math. univ. carolin., 22, 4, 761-772, (1981) · Zbl 0497.54038
[5] Antonyan, S.A.; Sanchis, M., Extension of locally pseudocompact group actions, Ann. mat. pura appl. (4), 181, 3, 239-246, (2002) · Zbl 1169.54356
[6] Arhangel’skii, A.V., Moscou spaces, pestov – tkachenko problem and C-embeddings, Comment. math. univ. carolin., 41, 3, 585-595, (2000) · Zbl 1038.54013
[7] Blasco, J.L.; Sanchis, M., On the product of two \(b_f\)-spaces, Acta math. acad. sci. hungar., 62, 111-118, (1993) · Zbl 0811.54015
[8] Bredon, Gl.E., Introduction to compact transformation groups, Pure and applied mathematics, vol. 46, (1972), Academic Press New York · Zbl 0246.57017
[9] Buchwalter, H., Produit topologique, produit tensoriel et c-repletion, Bull. soc. math. France suppl., Mém., 31-32, 51-71, (1972), (in French) · Zbl 0244.54012
[10] Comfort, W.W.; Hager, A.W., Uniform continuity in topological groups, (), 269-290
[11] van Douwen, E.K., βX and fixed-point free maps, Topology appl., 51, 2, 191-195, (1993) · Zbl 0792.54037
[12] Fay, T.H.; Ordman, E.T.; Smith-Thomas, B.V., The free group over rationals, Gen. topology appl., 10, 33-47, (1979) · Zbl 0403.22003
[13] Gillman, L.; Jerison, M., Rings of continuous functions, The university series in higher mathematics, (1960), D. Van Nostrand Princeton, NJ · Zbl 0093.30001
[14] Graev, M.I., Free topological groups, Amer. math. soc. transl., 8, 305-364, (1962)
[15] Hernández, S.; Sanchis, M.; Tkačenko, M., Bounded sets in spaces and topological groups, Topology appl., 101, 21-43, (2000) · Zbl 0979.54037
[16] Megrelishvili, M.; Scarr, T., Constructing Tychonoff G-spaces which are not G-Tychonoff, Topology appl., 86, 1, 69-81, (1998), (special issue on topological groups) · Zbl 0926.54027
[17] Palais, R.S., The classification of G-spaces, Mem. amer. math. soc. no., 36, (1960) · Zbl 0119.38403
[18] Sanchis, M., Continuous functions on locally pseudocompact groups, Topology appl., 86, 1, 5-23, (1998), (special issue on topological groups) · Zbl 0922.22001
[19] Tkachenko, M.G., Some properties of free topological groups, Math. notes, Mat. zametki, 37, 110-118, (1985), Russian original in: · Zbl 0568.22001
[20] de Vries, J., Glicksberg’s theorem for G-spaces, (), 663-673 · Zbl 0501.54027
[21] de Vries, J., On the G-compactification of products, Pacific J. math., 110, 2, 447-470, (1984) · Zbl 0487.54036
[22] de Vries, J., G-spaces: compactifications and pseudocompactness, (), 655-666 · Zbl 0598.54019
[23] de Vries, J., Equivariant embeddings of G-spaces, (), 485-493
[24] L. Waelbroek, Topological Vector Spaces and Algebras, Lectures at the Instituto de Matematica Pura y Aplicada, Rio de Janeiro, 1970
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.