Closed subsets of the domain whose image has the dimension of the range
The main theorem: Let f:\(X\to Y\) be a continuous map of a compact metric space X onto a compact metric space Y with \(\dim Y<\infty\) such that \(1\leq m\leq \dim f^{-1}(y)\leq n\) for all \(y\in Y\). Then there is a closed set \(K\subset X\) such that \(\dim K\leq n-m\) and \(\dim f(K)=\dim Y.\) Also, there is a closed set \(H\subset X\) such that \(\dim H\leq n-m\) and \(Int f(H)\neq \emptyset.\) This answers a question posed by J. Keesling and D. Wilson. It is noted that the result is best possible. The proof is rather complicated. Its technique comes from, as the author notes, \textit{J. L. Kelley} [Trans. Am. Math. Soc. 52, 22-36 (1942; Zbl 0061.40107)], \textit{L. Rubin}, \textit{R. M. Schori} and \textit{J. Walsh} [General Topol. Appl. 10, 93-102 (1979; Zbl 0413.54042)] and \textit{J. Keesling} and \textit{D. Wilson} [Topology Proc. 7, 91-107 (1982; Zbl 0541.54043)]. It is to be noted that this kind of results stems from an old problem due to Eilenberg: If \(\dim X>\dim Y>0,\) must there be a closed set \(K\subset X\) with \(\dim K<\dim f(K)?\) The problem is settled negatively by \textit{J. Keesling} and \textit{D. Wilson} [Proc. Am. Math. Soc. 86, 159-162 (1982; Zbl 0491.54030)]. The present main theorem gives a sufficient condition for a positive solution of the problem.
scientific article; zbMATH DE number 3857959
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