Irreducible continua of type \(\lambda\) with almost unique hyperspace
For a given continuum \(X\), consider a family \({\mathcal F}(X)\) of continua \(Y\) such that: (i) no two distinct members of \({\mathcal F}(X)\) are homeomorphic, (ii) \(C(Y)\) is homeomorphic to \(C(X)\) for each member \(Y\) of \({\mathcal F}(X)\) (the symbol \(C(X)\) denotes the hyperspace of all subcontinua of \(X\) with the Hausdorff metric), (iii) \({\mathcal F}(X)\) is the maximal family satisfying conditions (i) and (ii). A continuum \(X\) has a unique hyperspace if the family \({\mathcal F}(X)\) consists of one element only, namely \(X\); a continuum \(X\) has almost unique hyperspace if \({\mathcal F}(X)\) is finite and consists of more than one element. A continuum \(X\) which is irreducible between two points \(p\) and \(q\) is said to be of type \(\lambda\) provided that there is a monotone mapping \(g: X \rightarrow [0,1]\) such that \(g(p)=0\), \(g(q)=1\) and int\(_X (g^{-1}(t))= \emptyset\) for each \(t \in [0,1]\); the sets \(g^{-1}(t)\) are called layers of \(X\). This paper generalizes the results of \textit{G. Acosta} [Topology Appl. 117, No. 2, 175-189 (2002; Zbl 1003.54016)] and \textit{A. Illanes} [Topology Appl. 98, No. 1-3, 211-216 (1999; Zbl 0964.54004)]. The main result: if \(X\) is a continuum of type \(\lambda\) , each layer of \(X\) is a layer of cohesion and the set of degenerate layers is dense in \(X\), then: (a) if both end layers of \(X\) are nondegenerate, then \(X\) has unique hyperspace; (b) if exactly one end layer is degenerate, then \(X\) has either unique hyperspace or almost unique hyperspace. An example is constructed of an arc-like continuum \(X\) of type \(\lambda\) such that both end layers are degenerate and the family \({\mathcal F}(X)\) consists of infinitely many elements which are arc-like, too; this is a positive answer to a question of \textit{A. Illanes} [loc. cit.].
scientific article; zbMATH DE number 1744052
scientific article