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Homotopy separators and mappings into cubes. (English) Zbl 0679.55002

This is a continuation of the author’s previous work [Banach Cent. Publ. 18, 377-406 (1986; Zbl 0648.55013)].
If X is a space and f is a map of X into a product of compact manifolds (with boundary), then certain subsets of X, called homotopy separators of f, are defined. It is shown that maps into products of cells are in a sense distinguished among maps into products of arbitrary manifolds by a special property of the collection of their separators. These results are applied in estimating the dimension of certain subspaces of cubes. For instance, the following corollary is obtained: Let p: \(I^{n+1}\to I\) be the projection of an \((n+1)\)-cube onto its first factor. Then there exists a continuum \(A\subset I^{n+1}\) such that \(p(A)=I\), dim A\(=n\), and for any subset Y of A whose projection contains a copy of the Cantor set, dim Y\(=n\).
Reviewer: J.Jawarowski

MSC:

55M99 Classical topics in algebraic topology
54F45 Dimension theory in general topology
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
55M10 Dimension theory in algebraic topology

Citations:

Zbl 0648.55013
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