Inverse limits, inverse limit hulls and crossovers
The inverse limit space of an inverse sequence is a subset of the product of coordinate spaces. One may ask when is a subset of the product space the inverse limit of subsets of the factor spaces, i.e., if \(A\subset \Pi X_n\), are there subsets \(Y_n\subset X_n\) and maps \(f_n:Y_{n+1} \to Y_n\) such that \(A\) is the inverse limit of the inverse sequence \((Y_n, f_n)\)? Recently, mathematicians started to investigate situations when the bonding maps are set-valued functions and extended this question to such a situation. In the present paper the authors treat inverse sequences with compact metric coordinate spaces \(X_n\) and set-valued functions \(f_n: X_{n+1}\to 2^{X_n}\), where \(2^{X_n}\) denotes the set of closed subsets of \(X_n\). In order to recognize whether \(A\subset \Pi X_n\) is an inverse limit of subspaces of the coordinate spaces, \textit{V. Nall} [Houston J. Math. 37, No. 4, 1323--1332 (2011; Zbl 1237.54016)] introduced the notion of crossover for subsets \(A\subset \Pi X_n\), denoted \(Cr(A)\), and proved for closed \(A\) a result that characterizes the existence of an inverse sequence whose inverse limit is \(A\). Noticing that \(Cr\) is not idempotent, the authors of the present paper introduce two variants of Nall's crossover, denoted by \(CR(A)\) and \(\mathcal {CR}(A)\) and prove basic properties of all the three notions. Besides these variants they introduce the inverse limit hull \(\mathcal L\) of \(A\) as the smallest inverse limit containing \(A\). For closed \(A\subset \Pi X_n\) they prove a series of equivalent statements which state that there are closed subsets \(Y_n\subset X_n\) and u.s.c. functions \(f_n: Y_{n+1}\to 2^{Y_n}\) such that \(A\) is the inverse limit of \((Y_n,f_n)\).
upper semicontinuous set-valued functions
scientific article; zbMATH DE number 6509440