More on strong size properties (Q2787000)

Given a metric continuum \(X\), \(C_n (X)\) denotes the hyperspace of nonempty closed subsets of \(X\) having at most \(n\) components, \(C_n (X)\) is considered with the Hausdorff metric. Then, for \(n=1\), \(C_1 (X)\) is the hyperspace of subcontinua of \(X\).NEWLINENEWLINEA Whitney map for \(C_n (X)\) is a continuous function \({\mu} :C_n (X) {\rightarrow} [0,1]\) satisfying: (a) \({\mu}(X)=1\), (b) for each one-point set \(A\) in \(X\), \({\mu}(A)=0\) and (c) if \(A\) is a proper subset of \(B\), then \({\mu}(A)<{\mu}(B)\).NEWLINENEWLINEA Whitney level for \(C_n (X)\) is a set of the form \(\mu^{-1} (t)\), where \(0<t<1\).NEWLINENEWLINEIt is known that for \(n=1\) Whitney levels are connected, but this is not always true when \(n>1\). So, for \(n=1\), Whitney levels are subcontinua of \(C_1 (X)\) and it is possible to study them with all the tools developed in the area of Continuum Theory.NEWLINENEWLINELooking for some appropriate levels in \(C_n (X)\), for \(n>1\), \textit{H. Hosokawa} [Houston J. Math. 37, No. 3, 955--965 (2011; Zbl 1233.54004)] introduced the following definition. An strong size map is a continuous function \({\sigma} : C_n (X) {\rightarrow} [0,1]\) satisfying: (a') \({\sigma}(X)=1\), (b') for each finite element \(A\) of \(C_n (X)\), \({\sigma}(A)=0\) and (c') if \(A\) is a proper subset of \(B\) and \(B\) is not finite, then \({\mu}(A)<{\mu}(B)\). Hosokawa also proved that for each continuum \(X\) and each positive integer \(n\), \(C_n (X)\) there exist strong size maps and their fibers (strong size levels) are subcontinua of \(C_n (X)\).NEWLINENEWLINEThere have been four papers on strong size maps, namely: Hosokawa's paper [loc. cit.]; [\textit{S. Macías} and \textit{C. Piceno}, Glas. Mat., III. Ser. 48, No. 1, 103--114 (2013; Zbl 1275.54007)]; [\textit{L. Paredes-Rivas} and \textit{P. Pellicer-Covarrubias}, Topology Appl. 160, No. 13, 18116--1828 (2013; Zbl 1285.54004)] and the paper under review. The main purpose of them has been the study of the possible extensions, to strong size levels, of concepts, ideas, results and proofs related to Whitney levels.NEWLINENEWLINEIn particular, in the paper under review, the authors extend results related to admissible Whitney maps, Whitney preserving maps, and reversible Whitney properties. They prove that each of the following properties: being a continuum chainable continuum, being a locally connected continuum, and being a continuum with the property of Kelley is a reversible strong size property.