Cantor sets as generalized inverse limits (Q6614528)

Given a compact metric continuum \(X\) and an upper semi-continuous function \(F\) from \(X\) to the hyperspace of nonempty closed subsets of \(X\), the generalized inverse limit, \(\varprojlim F\), (GIL) is the space\N\[\N\varprojlim F=\{(x_{0},x_{1},\ldots)\in X^{\infty}:x_{i-1}\in F(x_{i})\text{ for all }i\in\mathbb{N}\}.\N\]\NGiven \(m\geq 2\), the partial \(m\)-GIL is the space\N\[\N\mathcal{G}_{m}(F)=\{(x_{0},\ldots,x_{m})\in X^{m}:x_{i-1}\in F(x_{i})\text{ and }1\leq i < m\}.\N\]\NSince \(\varprojlim F\) is, in fact, the limit of the sets \(\mathcal{G}_{m}(F)\), properties of these sets have been proved to be very useful in the study of properties of \(\varprojlim F\).\N\NA natural problem in this area is: given a compact metric space \(Z\), what conditions a function \(F\) must satisfy in order that \(\varprojlim F\) is homeomorphic to \(Z\)? For the case where \(Z\) is the Cantor set, this problem was studied in [\textit{F. Capulín} et al., Bol. Soc. Mat. Mex., III. Ser. 30, No. 3, Paper No. 78, 12 p. (2024; Zbl 07911000); Topol. Proc. 60, 71--80 (2022; Zbl 1487.54050)]. In both papers the role of the properties of the set \(D(F)=D_{1}(F)\cap D_{2}(F)\cap\cdots\), where \(D_{m}(F)=\pi_{1}\mathcal{(G}_{m}(F))\) and \(\pi_{1}\) is the projection on the first coordinate, played a fundamental role.\N\NIn the paper under review the authors characterize, in terms of \(D(F)\), the functions \(F\) for which \(\varprojlim F\) is a Cantor set. They divide the proof of their result in three cases: (1) \(D(F)\) is finite, in this case, the characterization is concise and easy to apply; (2) \(D(F)\) is countably infinite, in this case, some conditions are more difficult to identify but the authors illustrate the strength of this result with numerous examples; and (3) \(D(F)\) is uncountable.