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A method for constructing examples of M-equivalent spaces. (English) Zbl 0707.54007
Two Tychonoff spaces are called M-equivalent if their free topological groups in the sense of Markov are topologically isomorphic. Retractions \(r_ 1\) and \(r_ 2\) are parallel if \(r_ 1r_ 2=r_ 1\) and \(r_ 2r_ 1=r_ 2\). Images of the space X under parallel retractions are called parallel retracts of X. The author uses the notions of R-quotient mapping and space [S. M. Karnik and S. Willard, Can. Math. Bull. 25, 456-461 (1982; Zbl 0443.54020)]. He shows that if \(K_ 1\) and \(K_ 2\) are parallel retracts of a space X, then the R-quotient spaces \(X/K_ 1\) and \(X/K_ 2\) are M-equivalent. This allows to get a row of examples of pairs of M-equivalent spaces with different topological properties.
For instance Corollary 3.23: There exist M-equivalent spaces X and Y such that X is a first-countable locally compact space and Y is not a b-space and the tightness of Y is uncountable. In particular, the following properties are not preserved by M-equivalence: bisequentiality, the Fréchet-Urysohn property, sequentiality, k-property, countability of tightness.
Reviewer: L.B.Nachmanson

54B15 Quotient spaces, decompositions in general topology
22A05 Structure of general topological groups
54C15 Retraction
54G20 Counterexamples in general topology
54D50 \(k\)-spaces
54D55 Sequential spaces
Full Text: DOI
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