Abstract
We present several examples of feebly compact Hausdorff paratopological groups (i.e., groups with continuous multiplication) which provide answers to a number of questions posed in the literature. It turns out that a 2-pseudocompact, feebly compact Hausdorff paratopological group \(G\) can fail to be a topological group. Our group \(G\) has the Baire property, is Fréchet–Urysohn, but it is not precompact. It is well known that every infinite pseudocompact topological group contains a countable non-closed subset. We construct an infinite feebly compact Hausdorff paratopological group \(G\) all countable subsets of which are closed. Another peculiarity of the group \(G\) is that it contains a nonempty open subsemigroup \(C\) such that \(C^{-1}\) is closed and discrete, i.e., the inversion in \(G\) is extremely discontinuous. We also prove that for every continuous real-valued function \(g\) on a feebly compact paratopological group \(G\), one can find a continuous homomorphism \(\varphi \) of \(G\) onto a second countable Hausdorff topological group \(H\) and a continuous real-valued function \(h\) on \(H\) such that \(g=h\circ \varphi \). In particular, every feebly compact paratopological group is \(\mathbb{R }_3\) -factorizable. This generalizes a theorem of Comfort and Ross established in 1966 for real-valued functions on pseudocompact topological groups.
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Acknowledgments
The authors are grateful to O. Ravsky for a number of corrections and useful comments about the first version of the article.
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Communicated by A. Constantin.
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Sanchis, M., Tkachenko, M. Feebly compact paratopological groups and real-valued functions. Monatsh Math 168, 579–597 (2012). https://doi.org/10.1007/s00605-012-0444-3
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DOI: https://doi.org/10.1007/s00605-012-0444-3
Keywords
- Paratopological group
- Feebly compact
- 2-Pseudocompact
- Precompact
- Regularization
- \(\mathbb{R }_3\)-factorizable