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An inverse theorem on ’economic’ maps. (English. Russian original) Zbl 1253.54029

Sb. Math. 203, No. 4, 554-568 (2012); translation from Mat. Sb. 203, No. 4, 103-118 (2012).
The authors deal with ‘economic’ maps to Euclidean spaces; the quotes are apt as the notion is left mostly undefined. The reader can get an idea of what such a map might be by considering the main results of this paper; these are of the form: if \(\dim X\leq n\) then for most maps from \(X\) to \(\mathbb{R}^m\) and certain numbers \(d\) the preimages of \(d\)-dimensional hyperplanes are small in some sense. Various theorems of this kind can be found in [S. A. Bogatyi and V. M. Valov, Sb. Math. 196, No. 11, 1585–1603 (2005); translation from Mat. Sb. 196, No. 11, 33–52 (2005; Zbl 1141.54014)]. In this paper the authors prove that certain bounds in the latter paper are sharp.
Reviewer: K. P. Hart (Delft)

MSC:

54F45 Dimension theory in general topology
54C25 Embedding
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)

Citations:

Zbl 1141.54014
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