Constructing Tychonoff G -spaces which are not G -Tychonoff
Identifieur interne : 002137 ( Main/Exploration ); précédent : 002136; suivant : 002138Constructing Tychonoff G -spaces which are not G -Tychonoff
Auteurs : Michael Megrelishvili Levy [Israël] ; Tzvi Scarr [Israël]Source :
- Topology and its Applications [ 0166-8641 ] ; 1996.
Abstract
Jan de Vries' compactification problem is whether every Tychonoff G-space can be equivariantly embedded in a compact G-space. In such a case, we say that G is a V-group. De Vries showed that every locally compact group G is a V-group. The first example of a non-V-group was constructed in 1988 by the first author. Until now, this was the only known counterexample. In this paper, we give a systematic method of constructing noncompactifiable G-spaces. We show that the class of non-V-groups is large and contains all second countable (even ℵ0-bounded) nonlocally precompact groups. This establishes the existence of monothetic (even cyclic) non-V-groups, answering a question of the first author. As a related result, we obtain a characterization of locally compact groups in terms of “G-normality”.
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DOI: 10.1016/S0166-8641(97)00125-9
Affiliations:
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<front><div type="abstract" xml:lang="en">Jan de Vries' compactification problem is whether every Tychonoff G-space can be equivariantly embedded in a compact G-space. In such a case, we say that G is a V-group. De Vries showed that every locally compact group G is a V-group. The first example of a non-V-group was constructed in 1988 by the first author. Until now, this was the only known counterexample. In this paper, we give a systematic method of constructing noncompactifiable G-spaces. We show that the class of non-V-groups is large and contains all second countable (even ℵ0-bounded) nonlocally precompact groups. This establishes the existence of monothetic (even cyclic) non-V-groups, answering a question of the first author. As a related result, we obtain a characterization of locally compact groups in terms of “G-normality”.</div>
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