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Homogeneous subspaces of products of extremally disconnected spaces. (English) Zbl 07286530
In this exceptional paper the author establishes several results that extend or relate to results of Arhangel’skii, Frolík, Kunen, and others concerning homogeneity, products, extremally disconnected spaces (and the more general notion of a \(\beta\omega\)-space), and subspaces thereof. Maximal homogeneity, countable compactness, and pseudocompactness are also studied.
A. V. Arkhangel’skij [C. R. Acad. Sci., Paris, Sér. A 265, 822–825 (1967; Zbl 0168.43702)] proved that any compact subset of an extremally disconnected topological group is finite. In addition, Z. Frolík proved in [Commentat. Math. Univ. Carol. 8, 757–763 (1967; Zbl 0163.44503)] that homogeneous extremally disconnected compact spaces are finite. In the present paper the author obtains a simultaneous generalization of these two theorems by showing that all compact subsets of homogeneous extremally disconnected spaces are finite. In fact this result can be strengthened.
The main results include:
1. Any compact subspace of an homogeneous \(\beta\omega\)-space is finite (this is more general than what is mentioned in the abstract),
2. Under CH, if \(Y\) is a \(\beta\omega\)-space and \(X\subseteq Y^\omega\) is homogeneous, then every compact subspace of \(X\) is metrizable,
3. Under CH, if \(Y\) is a \(\beta\omega\)-space , and \(X\) is a homogeneous subspace of \(Y^n\) for some \(n\in\omega\), then any compact subspace of \(X\) is finite,
4. Let \(Y\) be an \(F\)-space, and let \(X\subseteq Y^3\) be a homogeneous space. Then any compact subspace of \(X\) is finite, and
5. If \(Y\) is an \(F\)-space, \(n\in\omega\), and \(X\subseteq Y^n\) is a homogeneous compact space, then \(X\) is finite.
The Keisler-Rudin order is used heavily in these results, as well as extensive notation introduced by the author involving sequences, sequences of open sets, and \(p\)-limits.
It is asked in Question 1 whether CH can be removed from 2 or 3 above, perhaps under the stronger assumptions that \(Y\) is an \(F\)-space or an extremally disconnected space. Question 2 is as follows: Let \(Y\) be an \(F\)-space, and let \(X\subseteq Y^4\) be a homogeneous space. Is it true that any compact subspace of \(X\) is finite?
MSC:
54B10 Product spaces in general topology
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
54B05 Subspaces in general topology
54E99 Topological spaces with richer structures
54H99 Connections of general topology with other structures, applications
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[1] Comfort, W. W.; Ross, K. A., Pseudocompactness and uniform continuity in topological groups, Pac. J. Math., 16, 3, 483-496 (1966) · Zbl 0214.28502
[2] Comfort, W.; Mill, J. V., On the product of homogeneous spaces, Topol. Appl., 21, 3, 297-308 (1985) · Zbl 0569.54011
[3] Comfort, W.; van Mill, J., A homogeneous extremally disconnected countably compact space, Topol. Appl., 25, 1, 65-73 (1987) · Zbl 0613.54027
[4] Lindgren, W. F.; Szymanski, A. A., A non-pseudocompact product of countably compact spaces via seq, Proc. Am. Math. Soc., 125, 12, 3741-3746 (1997) · Zbl 0891.54010
[5] Kato, A., A new construction of extremally disconnected topologies, Topol. Appl., 58, 1, 1-16 (1994) · Zbl 0804.54030
[6] Arhangelskii, A., Groupes topologiques extremalement discontinus, C. R. Acad. Sci. Paris, 265, 822-825 (1967) · Zbl 0168.43702
[7] Frolík, Z., Homogeneity problems for extremally disconnected spaces, Comment. Math. Univ. Carol., 008, 4, 757-763 (1967) · Zbl 0163.44503
[8] Arhangel’skii, A.; Tkachenko, M., Topological Groups and Related Structures (2008), Atlantis Press · Zbl 1323.22001
[9] van Douwen, E. K., Homogeneity of βg if g is a topological group, Colloq. Math., 41, 193-199 (1979) · Zbl 0454.22001
[10] Gillman, L.; Jerison, M., Rings of Continuous Functions (1960), Springer: Springer New York · Zbl 0093.30001
[11] van Douwen, E. K., Prime Mappings, Number of Factors and Binary Operations (1981), Instytut Matematyczny Polskiej Akademi Nauk · Zbl 0511.54013
[12] Balcar, B.; Dow, A., Dynamical systems on compact extremally disconnected spaces, Topol. Appl., 41, 1, 41-56 (1991) · Zbl 0766.54037
[13] Comfort, W. W.; Negrepontis, S., The Theory of Ultrafilters (1974), Springer Berlin Heidelberg · Zbl 0298.02004
[14] Kunen, K., Large homogeneous compact spaces, (van Mill, J.; Reed, G., Open Problems in Topology (1990), Elsevier Science Publishers B.V.: Elsevier Science Publishers B.V. North-Holland), 261-270
[15] Simon, P., Applications of independent linked families, Colloq. Math. Soc. János Bolyai, 41, 561-580 (1985) · Zbl 0615.54004
[16] Kunen, K., Weak p-points in \(n^\ast \), Colloq. Math. Soc. János Bolyai, 23, 741-749 (1978)
[17] Shelah, S.; Rudin, M. E., Unordered types of ultrafilters, Proceedings of the 1978 Topology Conference, I. Proceedings of the 1978 Topology Conference, I, Norman, OK, 1978. Proceedings of the 1978 Topology Conference, I. Proceedings of the 1978 Topology Conference, I, Norman, OK, 1978, Topol. Proc., 3, 1, 199-204 (1978), MR:540490, Zbl:0431.03033 · Zbl 0431.03033
[18] Rudin, W., Homogeneity problems in the theory of Čech compactifications, Duke Math. J., 23, 409-420 (1956) · Zbl 0073.39602
[19] Glicksberg, I., Stone-Čech compactifications of products, (The Mathematical Legacy of Eduard Čech (1993), Birkhäuser Basel), 67-80
[20] Ginsburg, J.; Saks, V., Some applications of ultrafilters in topology, Pac. J. Math., 57, 2, 403-418 (1975) · Zbl 0288.54020
[21] Reznichenko, E., Homogeneous products of spaces, Mosc. Univ. Math. Bull., 51, 3, 6-8 (1996) · Zbl 0908.54005
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