Inverse limits, inverse limit hulls and crossovers
type: Item
instance of: scholarly article
title: Inverse limits, inverse limit hulls and crossovers
zbMATH Open document ID: 1332.54160
author: Iztok Banič
author: Matevž Črepnjak
author: Matej Merhar
author: Uroš Milutinović
published in: Topology and its Applications
publication date: 2015-11-16T00:00:00Z
review text: 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 \((Y_n,f_n)\).
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54D80
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54F65
Mathematics Subject Classification ID: 54B99
zbMATH DE Number: 6509440
zbMATH Keywords: inverse limits
zbMATH Keywords: inverse limit hulls
zbMATH Keywords: upper semicontinuous set-valued functions
reviewed by: Ivan Ivanšić
MaRDI profile type: Publication
full work available at URL: j.topol.2015.09.040
OpenAlex ID: W2180345063
cites work: An Introduction to Inverse Limits with Set-valued Functions
cites work: Q5479987
cites work: Inverse limits with subsets of \([0,1]\times[0,1]\)
cites work: Q4004123
cites work: Q3115801
DOI: 10.1016/J.TOPOL.2015.09.040
description: scientific article; zbMATH DE number 6509440