The hyperspace of regular subcontinua (Q1688236)

Let \(X\) be a metric continuum. A subcontinuum \(A\) of \(X\) is regular closed if \(A\) is the closure of its interior. Denote by \(C(X)\) the hyperspace of subcontinua of \(X\), considered with the Hausdorff metric. The purpose of this paper is to study the subspace \(D(X)\) of \(C(X)\) consisting of all regular closed subcontinua of \(X\). The author mainly studies the connectedness and compactness of \(D(X)\). Among other results obtained in this paper, the author shows that: (a) if \(X\) is locally connected, then \(D(X)\) is locally connected, contractible and a dense subset of \(C(X)\); (b) there exists a dendroid \(X\) such that \(D(X)\) is disconnected; (c) for a continuum \(X\), \(D(X)\) is compact if and only if \(D(X)\) is finite; (d) there exists a decomposable continuum such that \(D(X)={X}\). The author says that Lemma 2.1 is easy to prove. In fact it easy to find a counterexample. This lemma holds for some special cases. In particular it holds when the metric is convex. It seems to be that the author only uses this lemma for these metrics. The paper finishes with the following question (Question 5.9). Does there exist a hereditarily decomposable continuum \(X\) for which \(D(X)={X}\)?