On Bing points in infinite-dimensional hereditarily indecomposable continua
The main thrust of this research is to show the existence of examples of infinite-dimensional continua with certain properties. Let \(X\) be a continuum (compact, connected, metrizable space). The set \(B_\infty(X)\) of Bing points of \(X\) consists of all \(x\) having the property that any non-trivial subcontinuum \(K\) of \(X\) containing \(x\) must satisfy \(\dim K= \dim X\). The second author and \textit{R. Pol} already have studied such sets in the case that \(dim X<\infty\) [Trans. Am. Math. Soc. 354, 2921-2932 (2002; Zbl 0992.54030)]. Now the case that dim\(X=\infty\) is considered. Techniques used in this paper may be associated with D. Henderson, M. Levin, and others. But, in particular, a method due to T. Maćkoviak is employed in a strong manner. This enables one, for example, to create a space and a map of it to the pseudoarc, whose fibers consist of interesting continua all of whose points are limit points of the domain (see Theorem 3.1). Some of the results of the paper are as follows. In section 4 (Theorem 4.6) it is proved that for each countable infinite ordinal \(\alpha\), there exist hereditarily indecomposable continua \(K\), \(L\) with \(\operatorname {Ind}K= \alpha=\operatorname {ind}L\). It is not demonstrated whether this could be done with \(K=L\), but the authors' Remark 4.7 shows that the technique they employ would not imply that. Examples in section 5 concern the existence of hereditarily indecomposable continua with certain properties. Let us state two of these. Assume again that \(\alpha\) is a countable infinite ordinal. Example 5.2. For any nonempty \(0\)-dimensional compactum \(D\), there exist hereditarily indecomposable continua \(X\) and \(Y\) with \(\operatorname {Ind}X=\alpha=\operatorname {ind}Y\) such that both \(B_\infty(X)\) and \(B_\infty(Y)\) are homeomorphic to \(D\). Example 5.5. There exists a hereditarily indecomposable continuum \(Z\) such that \(B_\infty(Z)\) is a \(1\)-dimensional dense \(G_\delta\)-subset of \(Z\) and \(\operatorname {Ind}Z=\alpha\) (the same could be done where \(\operatorname {ind}Z=\alpha\)). The previously described results dealt with countable-dimensional spaces, but in section 6 (the last one) the class of uncountable-dimensional spaces is considered. Here the authors make use of some techniques and theory developed by \textit{M. Levin} [Topology Appl. 65, 97-99 (1995; Zbl 0828.54028)]. An important development from the section is Example 6.4. There exists a weakly infinite-dimensional hereditarily indecomposable continuum \(M\) each composant of which is countable-dimensional (in fact, \(\sup\{\operatorname {ind}C:C\) is a composant of \(M\}<\omega_1)\) such that the set \(B_\infty(M)\) is strongly infinite-dimensional. Moreover, \(M\) is a C-space. This example was achieved in collaboration with R. Pol.
scientific article; zbMATH DE number 1879070
scientific article