Inverse limits of set-valued functions.
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Cornelius, Alexander Nelson.
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Much is known about inverse limits of compact spaces with continuous bonding maps. When the requirement that the bonding maps be continuous functions is relaxed, to allow for upper semi-continuous set-valued functions, many of the standard tools used to study inverse limit spaces are no longer available. Also, the structure of the inverse limit becomes more complicated. In this dissertation, the structure of compact subsets of the inverse limit is studied. Necessary and sufficient conditions are given for a compact subset of an inverse limit to be the inverse limit of its projections. Weak crossovers are introduced and used to describe the relationship between a set and the inverse limit of its projections. In the process of examining weak crossovers, tools with potentially broader applications are developed. When set-valued bonding maps are continuous and can be written as a union of continuous functions, sufficient conditions to make the inverse limit a Hausdorff continuum are given. It is also shown that, with another assumption, the inverse limit is in fact decomposable. When the bonding map can be written as a union of set-valued functions between compact subsets of the factor space, a description of the inverse limit is given. This uses a generalization of the itinerary of a point.