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Journal of Lie Theory 18 (2008), No. 4, 915--917
Copyright Heldermann Verlag 2008

A Counterexample in the Dimension Theory of Homogeneous Spaces of Locally Compact Groups

Adel A. George Michael
Mathematics and Sciences Unit, Dhofar University, P.O. Box 2509, 211 Salalah, Sultanate of Oman


We construct a locally compact group $G$ and a closed subgroup $H$ such that such that the quotient space $G/H$ is connected and has weight $w(G/H)=2^{\aleph_0}$ but fails to contain a cube $\I^{w(G/H)}$ of the same weight. This proves as incorrect an assertion made in Theorem 4.2 of K. H. Hofmann and S. A. Morris: Transitive actions of compact groups and topological dimension, J. of Algebra {\boldface 234} (2000), 454--479.

Keywords: Homogeneous spaces of locally compact groups, Tychonoff cube, dimension.

MSC: 22D05

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