Topology Proceedings 33 (2009), pp. 13-28

Some aspects of topological algebra and remainders of topological groups

A.V. Arhangel'skii

In the Introduction a very brief survey of some classical results and ideas of topological algebra is given. The principal interest in the article is directed at remainders of topological groups; here some new results are obtained. Thus, we continue the research line adopted in [A.V. Arhangel'skii, Remainders in compactifications and generalized metrizability properties, More on remainders close to metrizable spaces, On first countable remainders, Two types os remainders of topological groups]. Several results from these articles are improved. It is established that if a remainder of a non-locally compact topological group G is the union of a finite collection of metrizable spaces, then G is metrizable. A far reaching generalization of this result is also given; it is based on the notion of a D-space. If X is an uncountable Tychonoff space, and bY is a Hausdorff compactification of the space Y=C_p(X) such that the remainder bY \Y is homogeneous, then bY can be mapped continuously onto the Tychonoff cube I^w₁. Some further results and open problems on remainders of topological groups are provided.

Keywords: remainder, compactification, topological group, semitopological group, paratopological group, p-space, D-space, homogeneous space, metrizability, countable type, p-character, p-base

Mathematics Subject Classification: 54A25 (54B05)

Document formats: PDF file 154.1 Kb

Topology Proceedings, Volume 33 (2009)
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