More on Whitney levels of some decomposable continua (Q6633953)

In [\textit{E. Matsuhashi} and \textit{Y. Oshima}, Topology Appl. 326, Article ID 108395, 9 p. (2023; Zbl 1511.54019)] it was shown that being \(D\)-continua, \(D^*\)-continua, \(D^{**}\)-continua and Wilder continua are Whitney properties. The authors also posed the question of whether these properties are Whitney reversible (strong Whitney reversible, sequential strong Whitney reversible) properties.\N\NThis paper addressed these questions by constructing a non-\(D\)-continua \(Y\) such that each Whitney level of \(C(Y)\) is both \(D\)-continuum and a Wilder continuum. Additionally, it is proved that the property of being continuum-wise Wilder is not a Whitney property, while it is a sequential strong Whitney reversible property. Also, the authors exhibits a continuum-wise Wilder continuum \(X\) such that \(X \times [0,1]\) is not continuum-wise Wilder.\N\NRegarding the notion closed set-wise Wilder continua, introduced in this paper, the authors prove that the Cartesian product of two closed set-wise Wilder continua is closed set-Wilder, and the property of being closed set-wise Wilder continua is a sequential strong Whitney property.