Smoothness of hyperspaces and of Cartesian products (Q2701842)

Given a metric continuum \(X\) with a metric \(d\) and a point \(p\in X\), the symbols \(2^X\), \(C(X)\) and \(C(p,X)\) mean the hyperspace of all nonempty compact subsets of \(X\), of all nonempty subcontinua of \(X\), and of all nonempty subcontinua of \(X\) containing the point \(p\), respectively, all equipped with the Hausdorff metric \(H\). A ontinuum \(X\) is said to be smooth provided that there is a point \(p\in X\) such that for each \(\varepsilon>0\) there is a \(\delta> 0\) such that for each \(x\in X\), for each continuum \(K\in C(p,X)\cap C(x,X)\) and for each \(y\in X\) satisfying \(d(x,y) <\delta\) there is \(L\in C(p,X)\cap C(y,X)\) with \(H (K,L) <\varepsilon\). A continuum \(X\) is said to have the property of Kelley provided that for \(\varepsilon>0\) there is a \(\delta>0\) such that for each point \(x\in X\), for each \(K\in C(x,X)\) and for each point \(y\in X\) satisfying \(d(x,y) <\delta\) there is \(L\in C(y,X)\) with \(H(K,L) <\varepsilon\). The following results are shown:NEWLINENEWLINENEWLINETheorem 1. If the hyperspace \(2^X\) or \(C(X)\) of a continuum \(X\) is smooth, then \(X\) has the property of Kelley.NEWLINENEWLINENEWLINETheorem 2. If the Cartesian product \(X\times Y\) of nondegenerate continua \(X\) and \(Y\) is smooth, then each of the continua \(X\) and \(Y\) has the property of Kelley.NEWLINENEWLINENEWLINEExample 3. There is a continuum \(X\) having the property of Kelley such that \(2^X\) and \(X\times X\) are not smooth.NEWLINENEWLINENEWLINEA question is asked if \(C(X)\) is smooth when \(X\) has the property of Kelley.