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Two types of remainders of topological groups. (English) Zbl 1212.54086
Summary: We prove a dichotomy theorem: For each Hausdorff compactification \(bG\) of an arbitrary topological group \(G\), the remainder \(bG\setminus G\) is either pseudocompact or Lindelöf. It follows that, if a remainder of a topological group is paracompact or Dieudonné complete, then the remainder is Lindelöf, and the group is a paracompact \(p\)-space. This answers a question in A. V. Arkhangel’skij [Mosc. Univ. Math. Bull. 54, No. 3, 1–6 (1999); translation from Vestn. Mosk. Univ., Ser. I 1999, No. 3, 4–10 (1999; Zbl 0949.54054)]. It is shown that every Tychonoff space can be embedded as a closed subspace in a pseudocompact remainder of some topological group. We also establish some other results and present some examples and questions.

54D40 Remainders in general topology
54H11 Topological groups (topological aspects)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54C25 Embedding
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