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Maps improving the properties of spaces. (English. Russian original) Zbl 0849.54008
Russ. Math. Surv. 48, No. 1, 191-192 (1993); translation from Usp. Mat. Nauk. 48, No. 1(289), 187-188 (1993).
A map \(f: X\to Y\) is called an \(M\)-map if there is a continuous map \(F: X^3\to Y\) for which \(F(x, y, y)= F(y, y, x)= f(x)\) for any \(x,y\in X\). If the map \(f: X\to Y\) can be represented in the form of the composition of two maps \(g: X\to Z\) and \(h: Z\to Y\), where \(Z\) is a topological group (\(M\)-space), then \(f\) is an \(M\)-map. In particular, any continuous map of a topological group onto an arbitrary space is an \(M\)-map.
Theorem. Let \(f: \Pi\to X\) be a continuous \(M\)-map of the product \(\Pi\) of Lindelöf \(\Sigma\)-spaces onto \(X\). Then (a) each uncountable regular cardinal is a weak precalibre of the space \(X\) (therefore \(c(X)\leq \aleph_0\)); (b) the space \(X\) is \(\tau\)-cellular for each cardinal \(\tau\geq \aleph_0\).

MSC:
54C05 Continuous maps
54H15 Transformation groups and semigroups (topological aspects)
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