On inverse limits with set-valued functions on graphs, dimensionally stepwise spaces and ANRs

A space X is a dimensionally stepwise space if dim⁡X<∞ and for any 1≤m≤dim⁡X there is an open set Um of X such that dim⁡Um=m. In this paper, for given inverse sequence {Xi,fi,i+1}i=1∞ of compacta with upper semi-continuous set-valued functions, we introduce new indexes I˜({Xi,fi,i+1}) and W˜({Xi,fi,i+1}), and by use of the indexes we investigate topological structures of inverse limits of graphs with upper semi-continuous set-valued functions. Especially, we prove the following theorems.