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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.

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