✦ LIBER ✦

Constructing Tychonoff G-spaces which are not G-Tychonoff

✍ Scribed by Michael Megrelishvili (Levy); Tzvi Scarr

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 799 KB
Volume: 86
Category: Article
ISSN: 0166-8641
DOI: 10.1016/s0166-8641(97)00125-9

No coin nor oath required. For personal study only.

✦ Synopsis

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-L7-groups is large and contains all second countable (even No-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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