Some decomposable continua and Whitney levels of their hyperspaces

TL;DR

This paper introduces a new class of continua called $D^{**}$-continua, explores their relationships with existing classes, and shows that certain properties are Whitney properties, providing a negative answer to a previously posed question.

Contribution

The paper defines $D^{**}$-continua, compares them with Wilder, $D^*$, and $D$-continua, and proves that several classes are Whitney properties, advancing the understanding of hyperspace levels.

Findings

$D^{**}$-continua form a new class strictly larger than Wilder and $D^*$-continua.

The class of $D$-continua is larger than the class of $D^{**}$-continua.

Being Wilder, $D$, $D^*$, and $D^{**}$ are Whitney properties.

Abstract

We introduce the new class of continua; $D^{**}$-$continua$. The classes of Wilder continua and $D^{*}$-continua are strictly contained in the class of $D^{**}$-continua. Also, the class of $D$-continua is bigger than the class of $D^{**}$-continua. Using $D^{**}$-continua, we give the negative answer to a Question. Furthermore, we prove that being Wilder, being $D$, being $D^*$ and being $D^{**}$ are Whitney properties.