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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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