Homogeneity in powers

of subspaces of the real line

Author:
L. Brian Lawrence

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

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Abstract | References | Similar Articles | Additional Information

Abstract: Working in ZFC, we prove that for every zero-dimensional subspace of the real line, the Tychonoff power is homogeneous ( denotes the nonnegative integers). It then follows as a corollary that is homogeneous whenever 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.

**1.**A. Dow and E. Pearl,*Homogeneity in powers of first countable, zero-dimensional spaces*, Proc. Amer. Math. Soc. (to appear).**2.**F. van Engelen,*On the homogeneity of infinite products*, Topology Proc.**17**(1992), 303-315. MR**95b:54044****3.**R. Engelking,*General Topology*, Polish Scientific Publishers, 1977. MR**58:18316b****4.**G. Gruenhage, ed., Topology Proc.**15**(1990), 207-208.**5.**O. H. Keller,*Die homoiomorphie der Kompakten Konvexen Mengen im Hillbertschen Raum*, Math. Ann.**105**(1931), 748-758.**6.**S. V. Medvedev,*Characterizations of -homogeneous spaces*, Interim Report of the Prague Topological Symposium 2 (1987).**7.**J. van Mill,*A rigid space for which is homogeneous; an application of infinite-dimensional topology*, Proc. Amer. Math. Soc.**83**(1981), 597-600. MR**82h:54067****8.**J. van Mill and G. M. Reed, Eds.,*Open Problems in Topology*, North-Holland, Amsterdam, 1990. MR**92c:54001****9.**D. B. Motorov,*Homogeneity and -bases*, Vestnik Moscow Univ. Ser. 1 (1989) No. 4, 31-34. MR**90m:54049**

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Additional Information

**L. Brian Lawrence**

Affiliation:
Department of Mathematics, George Mason University, Fairfax, Virginia 22030-4444

Email:
blawrenc@osf1.gmu.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-02100-X

Keywords:
Real line,
separable metric space,
zero-dimensional,
subspace,
product space,
power,
homogeneous,
rigid

Received by editor(s):
September 7, 1994

Received by editor(s) in revised form:
June 1, 1995

Article copyright:
© Copyright 1998
American Mathematical Society