Authors: Spěvák, Jan, Department of mathematics, J.E. Purkinje University, Ceske mladeze 8, 400 96 Usti nad Labem, Czech Republik (jan.spevak@ujep.cz).
Finite-valued mappings preserving dimension, pp. 327-348.
ABSTRACT. We say that a set-valued mapping F: X⇒Y is C-lsc provided that there exists a countable cover C of X consisting of functionally closed sets such that for every C∈C and each functionally open subset U of Y one can find a functionally open set V⊂X such that {x∈C: F(x)∩ U≠Ø}=C∩V. For Tychonoff spaces X and Y we say that X dominates Y provided that there exist a finite-valued C-lsc mapping F: X⇒Y and a finite-valued D-lsc mapping G:Y⇒X (for suitable C and D) such that y∈ ∪{F(x):x∈G(y)} for every y∈Y. We prove that if X dominates Y, then dim X≥dim Y. (Here dim X denotes the Čech-Lebesgue (covering) dimension of X.) As a corollary, we obtain that dim X=dim Y whenever a perfectly normal space Y is an image of a Tychonoff space X under a finite-to-one open mapping. We also give an example of an open mapping f:X→Y such that |f^-1(y)|≤2 for all y∈Y, both X and Y are hereditarily normal (and Y is even Lindelöf) but dim X≠dim Y.