More on Whitney levels of some decomposable continua

In this paper, we show that there exists a non-D-continuum such that each positive Whitney level of the hyperspace of subcontinua of the continuum is both $D^{⁎}$ and Wilder. We show that the property of being continuum-wise Wilder is not a Whitney property, while it is a Whitney reversible property. Furthermore, we introduce the new class of continua: closed set-wise Wilder continua. This class is larger than the class of continuum chainable continua and smaller than the class of continuum-wise Wilder continua. In addition to the above results, we show that the Cartesian product of two closed set-wise Wilder continua is close set-wise Wilder.