The (weak) full projection property for inverse limits with upper semicontinuous bonding functions (Q1790549)

We start with a statement of the standard full projection property theorem for inverse limits: Suppose \(X\) is the inverse limit of an inverse sequence of compact metric spaces \(X_i\) with continuous single-valued surjective bonding maps. Let \(A\subset X\) be a closed subset. Assume that for infinitely many indices \(i\), the \(i\)th projection of \(A\) to \(X_i\) is surjective. Then \(A=X\). It is well known that this result does not hold in the general case if the bonding maps are upper semicontinuous, surjective, and set-valued. In this paper the authors prove various settings for upper semicontinuous and surjective set-valued bonding maps, in which a variant of this property holds. First, the authors distinguish the full projection property and the weak full projection property, the latter one requiring that for all indices \(i\) the \(i\)th projection of \(A\) to \(X_i\) is surjective. The concepts coincide if the bonding maps are single-valued. To each of these properties a chapter is dedicated in the paper. The results of the paper follow a series of questions on the topic stated by Ingram and solve some of them. They provide sufficient conditions on the bonding maps for the inverse limit to have the (weak) full projection property. Furthermore, they provide partial results towards characterization of the (weak) full projection property in terms of the bonding maps. The results usually relate the (weak) full projection property with (weakly) irreducible maps.