Parametric Bing and Krasinkiewicz maps
All spaces in this paper are assumed to be metrizable and all maps continuous. A function space \(C(X,M)\) is endowed with the source limitation topology. A compact space \(X\) is called a Bing space if each of its subcontinua is hereditariy indecomposable. A map \(g\) is said to be a Bing map if all fibers of \(g\) are Bing spaces. The space \(M\) is a free space if for any compact space \(X\) the function space \(C(X,M)\) contains a dense subset consisting of Bing maps. A map \(g:X\to M\) between compact spaces is called a Krasinkiewicz map if every continuum in \(X\) is either contained in a fiber of \(g\) or contains a component of a fiber of \(g\). The class of free spaces contains all \(n\)-manifolds [\textit{J. Krasinkiewicz}, Bull. Pol. Acad. Sci., Math. 44, No. 2, 147--156 (1996; Zbl 0867.54020)], all locally finite polyhedra, all manifolds modeled on Menger cubes or Nöbeling spaces and 1-dimensional locally connected continua [\textit{J. Song} and \textit{E. D. Tymchatyn}, Fundam. Math. 163, No. 3, 229--239 (2000; Zbl 0971.54013)]. \textit{J. Krasinkiewicz} [Bull. Pol. Acad. Sci., Math. 44, No. 4, 431--440 (1996; Zbl 0885.54009)] proved that, for any compact space \(X\), the Krasinkiewicz maps form a dense subset \(C(X,[0,1])\). \textit{M. Levin} and \textit{W. Lewis} [Isr. J. Math. 133, 61--76 (2003; Zbl 1031.54019)] established that this set is also a \(G_\delta\)-subset of \(C(X,[0,1])\). In the paper under review, the author gives parametric versions of these results by proving the following theorems. Theorem 1.1. Let \(f:X\to Y\) be a perfect map with \(Y\) being a strongly countable-dimensional space. Then, for every complete \(ANR\) free space \(M\), the function space \(C(X,M)\) contains a dense \(G_\delta\)-set of maps \(g\) such that all restrictions \(g| f^{ - 1}(y), y\in Y\), are Bing maps. Moreover, if in addition \(M\) is a closed convex subset of a Banach space, then the same conclusion remains true provided that \(Y\) is a \(C\)-space. Theorem 1.2. Let \(f:X\to Y\) be a perfect map with \(Y\) being a \(C\)-space. Then \(C(X, [0,1])\) contains a \(G_\delta\)-subset of maps such that all restrictions \(g| f^{ - 1}(y), y\in Y\), are Bing and Krasinkiewicz maps.
scientific article; zbMATH DE number 5275715
scientific article