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\(\mathbb{R}\)-factorizability of topological groups and \(G\)-spaces. (English) Zbl 1521.54022

The reviewed paper is (probably) written for researchers with a clear inclination to consider axiomatic theories with roots in general topology, in particular for researchers who want to specialize in some categorical aspects related to topological groups. The considerations concern lots of concepts and facts with which the author only briefly acquaints the readers, referring to the literature not only in the cited monographs. The meaning of some abbreviations must be guessed by the reader, for example an abbreviation “the category G-Tych” used more than ten times. For these reasons it is tedious to read, so instead of the author’s discussion, I am posting a slightly modified abstract of the paper.
It is investigated the relation between \(\mathbb R\)-factorizability of a G-space in the category G-Tych and \(\mathbb R\)-factorizability of its acting group. It is shown that \(\mathbb R\)-factorizability of an acting group with d-open action doesn’t imply the \(\mathbb R\)-factorizability of a G-space. Transitivity of a d-open action of an \(\mathbb R\)-factorizable group implies \(\mathbb R\)-factorizability of G-space in the category G-Tych. Moreover, if a d-openly acting group is openly factorizable, then the G-space is \(\mathbb R\)-factorizable in the category G-Tych. Also, a \(\sigma\)-lattice of open homomorphism on G induces a \(\sigma\)-lattice of equivariant d-open maps on \((G,X,\alpha)\) and \(X\) is an I-favorable space.

MSC:

54H11 Topological groups (topological aspects)
54H15 Transformation groups and semigroups (topological aspects)
54B35 Spectra in general topology
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