A non-D-continuum with weakly infinite-dimensional closed set-aposyndetic Whitney levels
Eiichi Matsuhashi
TL;DR
The paper addresses how dimensional-aposyndetic properties interact with Whitney levels in hyperspace topology by introducing weakly infinite-dimensional closed set-aposyndetic continua and constructing a non-$D$-continuum $Z$ whose positive Whitney levels $oldsymbol{ u}^{-1}(t)$ are weakly infinite-dimensional closed set-aposyndetic continua. The approach modifies a prior Illanes-type construction, employing a superdendrite $D$ and a tree-like continuum to ensure non-$D$ and to enforce aposyndesis across all Whitney levels. A key contribution is the explicit demonstration that every positive Whitney level of $Z$ has the weakly infinite-dimensional closed set-aposyndetic property, along with a finite-dimensional counterexample illustrating hierarchy separations among posyndesis notions. This work sharpens the understanding of how Whitney-level properties reflect in hyperspaces and strengthens earlier results by van Douwen, Illanes, and colleagues.
Abstract
In this paper, we introduce the new class of continua; weakly infinite-dimensional closed set-aposyndetic continua. With this notion, we show that there exists a non-D-continuum such that each positive Whitney level of the hyperspace of the continuum is a weakly infinite-dimensional closed set-aposyndetic continuum. This result strengthens those of van Douwen and Goodykoontz [2], Illanes [7], and the main result of Illanes et al. [9].