CONCERNING DIMENSION AND TREE-LIKENESS OF INVERSE LIMITS WITH SET-VALUED FUNCTIONS

From a theorem of Van Nall it is known that inverse limits with sequences of upper semi-continuous set-valued functions with 0-dimensional values have dimension bounded by the dimensions of the factor spaces. Information is also available about the dimension of inverse limits with sequences of upper semi-continuous continuum-valued bonding functions having graphs that are mappings on the factor spaces that have continua appended at each point of a closed set, however the conclusion of this theorem allows the possibility of an infinite dimensional inverse limit. In this paper we show that inverse limits with sequences of certain surjective upper semi-continuous continuum-valued bonding functions have dimension bounded by the dimensions of the factor spaces. One consequence of our investigation is that certain inverse limits on [0, 1] with upper semi-continuous continuum-valued functions are tree-like including those that are inverse limits on [0, 1] with a single interval-valued bonding function that has no flat spots.