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A note on splittable spaces. (English) Zbl 0769.54004
A space $$X$$ is splittable over a space $$Y$$ if for every $$A\subset X$$ there exists a continuous map $$f: X\to Y$$ with $$f^{-1} fA=A$$. The author proves that any $$n$$-dimensional polyhedron splits over $$\mathbb{R}^{2n}$$ but not necessarily over $$\mathbb{R}^{2n-2}$$. It is established that if a metrizable compact space $$X$$ splits over $$\mathbb{R}^ n$$, then $$\dim X\leq n$$. An example of $$n$$-dimensional compact space which does not split over $$\mathbb{R}^{2n}$$ is given.

##### MSC:
 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
##### Keywords:
splittability; polyhedron
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