Closed subsets of the domain whose image has the dimension of the range
type: Item
instance of: scholarly article
title: Closed subsets of the domain whose image has the dimension of the range
zbMATH Open document ID: 0539.54027
DOI: 10.1016/0166-8641(84)90021-X
author: Eiji Kurihara
published in: Topology and its Applications
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 3857959
zbMATH Keywords: proper mapping
zbMATH Keywords: dimensional inequalities
MaRDI profile type: Publication
cites work: Closed mappings which lower dimension
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description: scientific article; zbMATH DE number 3857959