Homogeneity in powers of subspaces of the real line
HTML articles powered by AMS MathViewer
- by L. Brian Lawrence PDF
- Trans. Amer. Math. Soc. 350 (1998), 3055-3064 Request permission
Abstract:
Working in ZFC, we prove that for every zero-dimensional subspace $S$ of the real line, the Tychonoff power ${}^\omega S$ is homogeneous ($\omega$ denotes the nonnegative integers). It then follows as a corollary that ${}^\omega S$ is homogeneous whenever $S$ is a separable zero-dimensional metrizable space. The question of homogeneity in powers of this type was first raised by Ben Fitzpatrick, and was subsequently popularized by Gary Gruenhage and Hao-xuan Zhou.References
- A. Dow and E. Pearl, Homogeneity in powers of first countable, zero-dimensional spaces, Proc. Amer. Math. Soc. (to appear).
- Fons van Engelen, On the homogeneity of infinite products, Topology Proc. 17 (1992), 303–315. MR 1255813
- Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
- G. Gruenhage, ed., Topology Proc. 15 (1990), 207–208.
- O. H. Keller, Die homoiomorphie der Kompakten Konvexen Mengen im Hillbertschen Raum, Math. Ann. 105 (1931), 748–758.
- S. V. Medvedev, Characterizations of $h$-homogeneous spaces, Interim Report of the Prague Topological Symposium 2 (1987).
- Jan van Mill, A rigid space $X$ for which $X\times X$ is homogeneous; an application of infinite-dimensional topology, Proc. Amer. Math. Soc. 83 (1981), no. 3, 597–600. MR 627701, DOI 10.1090/S0002-9939-1981-0627701-2
- Jan van Mill and George M. Reed (eds.), Open problems in topology, North-Holland Publishing Co., Amsterdam, 1990. MR 1078636
- D. B. Motorov, Homogeneity and $\pi$-networks, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4 (1989), 31–34, 105 (Russian); English transl., Moscow Univ. Math. Bull. 44 (1989), no. 4, 45–50. MR 1029749
Additional Information
- L. Brian Lawrence
- Affiliation: Department of Mathematics, George Mason University, Fairfax, Virginia 22030-4444
- Email: blawrenc@osf1.gmu.edu
- Received by editor(s): September 7, 1994
- Received by editor(s) in revised form: June 1, 1995
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 3055-3064
- MSC (1991): Primary 54B10; Secondary 54E35, 54F99
- DOI: https://doi.org/10.1090/S0002-9947-98-02100-X
- MathSciNet review: 1458308