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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)
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