# zbMATH — the first resource for mathematics

On power homogeneous spaces. (English) Zbl 0994.54028
A topological space $$X$$ is called power homogeneous if there exists a cardinal number $$\tau>0$$ such that the space $$X^\tau$$ is homogeneous. It may easily happen that a space $$X$$ is not homogeneous while $$X$$ is power homogeneous. Every first countable zero-dimensional Hausdorff space is $$\omega$$-power homogeneous. In this paper, some new necessary conditions for a space to be power homogeneous are obtained. The notions of a Moscow space and a weakly Klebanov space are applied to study power homogeneous spaces. In particular it is proved that (1) every Corson compact power homogeneous space is first countable; (2) a compact scattered space is power homogeneous if and only if it is countable.
Reviewer: Shou Lin (Fujian)

##### MSC:
 54D50 $$k$$-spaces 54D60 Realcompactness and realcompactification 54C35 Function spaces in general topology
Full Text:
##### References:
 [1] Amirdjanov, G.P., On dense subspaces of countable pseudocharacter and other generalizations of separability, Dokl. akad. nauk SSSR, 235, 5, 993-996, (1977) [2] Arhangel’skii, A.V., Functional tightness, Q-spaces, and τ-embeddings, Comment. math. univ. carolinae, 24, 1, 105-120, (1983) · Zbl 0528.54006 [3] Arhangel’skii, A.V., Topological homogeneity. topological groups and their continuous images, Russian math. surveys, 42, 2, 83-131, (1987) · Zbl 0642.54017 [4] Arhangel’skii, A.V., Topological function spaces, (1992), Kluwer Academic Dordrecht · Zbl 0911.54004 [5] Arhangel’skii, A.V., On a theorem of W.W. comfort and K.A. ross, Comment. math. univ. carolin., 40, 1, 133-151, (1999) [6] Arhangel’skii, A.V., Moscow spaces, pestov – tkačenko problem, and C-embeddings, Comment. math. univ. carolin., 41, (2000) · Zbl 1038.54013 [7] Bell, M., Nonhomogeneity of powers of COR images, Rocky mount. J. math., 22, 3, 805-812, (1992) · Zbl 0796.54032 [8] Comfort, W.W.; Ross, K.A., Pseudocompactness and uniform continuity in topological groups, Pacific J. math., 16, 3, 483-496, (1966) · Zbl 0214.28502 [9] van Douwen, E., Nonhomogeneity of products of preimages and π-weight, Proc. amer. math. soc., 69, 1, 183-192, (1978) · Zbl 0385.54004 [10] van Douwen, E., Homogeneity of βG if G is a topological group, Colloq. math., 41, 193-199, (1979) · Zbl 0454.22001 [11] E. van Douwen, T.C. Przymusiński, First countable and countable spaces all compactifications of which contain βN, Fund. Math. (102) 229-234 · Zbl 0398.54016 [12] Dow, A.; Pearl, E., Homogeneity in powers of zero-dimensional first countable spaces, Proc. amer. math. soc., 125, 2503-2510, (1997) · Zbl 0963.54002 [13] Engelking, R., General topology, (1977), PWN Warszawa [14] Frolik, Z., The topological product of two pseudocompact spaces, Czechoslovak math. J., 10, 339-349, (1960) · Zbl 0099.38503 [15] Gillman, L.; Jerison, M., Rings of continuous functions, (1960), Princeton Univ. Press Princeton, NJ · Zbl 0093.30001 [16] Glicksberg, I., Stone-čech compactifications of products, Trans. amer. math. soc., 90, 369-382, (1959) · Zbl 0089.38702 [17] Isbell, J.R., Zero-dimensional spaces, Tôhoku math. J., 7, 1-2, 3-8, (1955) · Zbl 0066.41103 [18] Klebanov, B.S., Remarks on subsets of Cartesian products of metric spaces, Comment. math. univ. carolin., 23, 767-784, (1982) · Zbl 0507.54011 [19] Kunen, K., Large homogeneous compact spaces, (), 261-270 [20] Malychin, V.I., On homogeneous spaces, (), 50-61 · Zbl 0801.54003 [21] Noble, N., The continuity of functions on Cartesian products, Trans. amer. math. soc., 149, 187-198, (1970) · Zbl 0229.54028 [22] Ohta, H.; Sakai, M.; Tamano, Ken-Ichi, Perfect κ-normality in product spaces, (), 279-289 · Zbl 0835.54009 [23] Reznichenko, E.A., Homogeneous products of spaces, Moscow univ. math. bull., 51, 3, 6-8, (1996) · Zbl 0908.54005 [24] Semadeni, Z., Sur LES ensembles clairsemés, Dissertationes math., 19, (1959), Warszawa [25] Ščepin, E.V., On κ-metrizable spaces, Izv. akad. nauk SSSR ser. mat., 43, 2, 442-478, (1979) · Zbl 0409.54040 [26] Tkačenko, M.G., The notion of o-tightness and C-embedded subspaces of products, Topology appl., 15, 93-98, (1983) · Zbl 0509.54013 [27] Tkačenko, M.G., Introduction to topological groups, Topology appl., 86, 3, 179-231, (1998) · Zbl 0955.54013 [28] Uspenskij, V.V., For any X, the product X×Y is homogeneous for some Y, Proc. amer. math. soc., 87, 1, 187-188, (1983) · Zbl 0504.54007 [29] Uspenskij, V.V., A large Fσ-discrete Fréchet space having the souslin property, Comment. math. univ. carolin., 25, 2, 257-260, (1984) · Zbl 0553.54001 [30] Uspenskij, V.V., Topological groups and dugundji spaces, Mat. sb., 180, 8, 1092-1118, (1989) · Zbl 0684.22001 [31] Yajima, Y., The normality of σ-products and the perfect κ-normality of Cartesian products, J. math. soc. Japan, 36, 4, 689-699, (1984) · Zbl 0556.54007
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.