Abstract
We show that if an extremally disconnected space X has a homogeneous compactification, then X is finite. It follows that if a totally bounded topological group has a dense extremally disconnected subspace, then it is finite. The techniques developed in this article also imply that if the square of a topological group G has a dense extremally disconnected subspace, then G is discrete. See also Theorem 3.12. We also establish a sufficient condition for an extremally disconnected topological ring to be discrete (Theorem 3.9). A theorem on the structure of an arbitrary homeomorphism of an extremally disconnected topological group onto itself is proved (see Theorem 3.7 and Corollary 3.8).
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Communicated by S.K. Jain.
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Arhangel’skii, A. A study of extremally disconnected topological spaces. Bull. Math. Sci. 1, 3–12 (2011). https://doi.org/10.1007/s13373-011-0001-8
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DOI: https://doi.org/10.1007/s13373-011-0001-8
Keywords
- Extremally disconnected
- Compactification
- Topological group
- Dyadic compactum
- Homogeneous space
- Totally bounded group