Models of hyperspaces (Q2850619)

The author discusses models of hyperspaces of metric continua; a model for a given hyperspace \(\mathcal K(X)\) of a metric continuum \(X\) is a topologically equivalent space, where the elements are points instead of subsets. The author considers the following hyperspaces and finds their models : \( 2^X=\{A\subset X:A\text{ is nonempty and closed in }X\}\), \(C_n(X)=\{A\in2^X:A\text{ has at most }n\text{ components}\}\), \(F_n(X)=\{A\in2^X:A\text{ has at most }n\text{ points}\}\), \(C(X)=C_1(X)\). Here \(X\) is a metric continuum, i.e. a nondegenerate compact connected metric space and all the hyperspaces are considered with the Hausdorff metric \(H\) defined as NEWLINE\[NEWLINEH(A,B)=\max\big\{\max\{d(a,B):a\in A\},\,\max\{d(b,A):b\in B\}\big\},NEWLINE\]NEWLINE where \(d(a,B)=\min\{d(a,b):b\in B\}\).NEWLINENEWLINEThe author considers a number of metric continua, for example, the unit interval [0,1] in \(\mathbb R\), the circle \(S^1\), the noose, the simple triod and more generally the simple \(n\)-od (\(n\geq3\)) in order to find the model of the corresponding hyperspaces they generate. In this context the author tries to find some characterisation of the locally connected continua \(X\) for which the aforesaid hyperspaces are embeddable in \(\mathbb R^n\) for some \(n\geq3\). Then the author cites infinite dimensional models for various hyperspaces; for example, \(2^X\) is homeomorphic to the Hilbert cube iff \(X\) is a locally connected continuum.NEWLINENEWLINEActually this paper is a survey work on models of various hyperspaces based on a number of papers by many people including the author. Throughout this paper a number of open problems regarding models and embeddings of hyperspaces are posed by the author.