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On mappings of compact spaces into Cartesian spaces

✍ Scribed by Semeon A. Bogatyi; Vitaly V. Fedorchuk; Jan van Mill

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
114 KB
Volume
107
Category
Article
ISSN
0166-8641

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✦ Synopsis

Eilenberg proved that if a compact space X admits a zero-dimensional map f :X β†’ Y , where Y is m-dimensional, then there exists a map h :X β†’ I m+1 such that f Γ— h :X β†’ Y Γ— I m+1 is an embedding. In this paper we prove generalizations of this result for Οƒ -compact subsets of arbitrary spaces. An example of a compact space X and of a zero-dimensional Οƒ -compact subset A βŠ‚ X is given such that for any continuous function f :X β†’ R which is one-to-one on the set A and any G Ξ΄ -subset B of X with B βŠƒ A the restriction f |B :B β†’ R has infinite fibers. This example is used to demonstrate that our results are sharp.

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