\(Z\)-sets in hyperspaces (Q2771259)

Let \((Y,d)\) be a metric space. A closed subset \(A\) of \(Y\) is a Z-set if for each \(\varepsilon >0\) there exists a continuous mapping \(f_\varepsilon\) of \(Y\) into \(Y\setminus A\) such that \(d(y,f_\varepsilon (y))<\varepsilon\) for all \(y\in Y\). The authors consider different hyperspaces (e.g. \(2^X\), the space of all closed subsets, \(\mathcal C (X)\), the space of all connected subsets, \(\mathcal C_n (X)\), the space of all subsets having at most \(n\) components, \(\mathcal F_n\), the space of all subsets consisting of at most \(n\) points) for different continua \(X\) (e.g. with the property of Kelley, homogeneous, hereditarily indecomposable, fans, smooth dendroids, arc-smooth in a point). They prove in different situations that one hyperspace is a \(Z\)-set in another. For example, if \(X\) is a continuum with the property of Kelley, then \(\mathcal F_n\) is a \(Z\)-set in \(2^X\) and in \(\mathcal C_n\) for any \(n\). There are many examples and a list of problems.