Topics in Generalised Inverse Limits PDF Download

Author: Michael Lockyer
Publisher:
ISBN:
Category : Continuum (Mathematics)
Languages : en
Pages : 111

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Book Description
Generalised inverse limits are a new topic of study in the area of continuum theory. Although similarly defined to the more traditional inverse limits of continuous functions on continua, they have a much richer structure. It was realised early on that many of the theorems relating to the inverse limits of continuous functions do not carry over to generalised inverse limits, which use set valued functions. Much of the research in generalised inverse limits to this point has been to attempt to characterise their structure. This thesis contains three main chapters, each of which is based on a topic in generalised inverse limits research. In the first of these we explore the structure of a particular generalised inverse limit known as K(0,1). This is the inverse limit of a generalised tent map. The inverse limits of tent maps have played an important role in continuum theory in the past, and with the introduction of generalised inverse limits we can include generalised tent maps that are not continuous functions. The only such generalised tent map whose inverse limit does not have a very simple structure is K(0,1). In this chapter we prove a number of topological properties of K(0,1), and give an embedding of K(0,1)) into R3. For the second topic we characterise a certain kind of disconnection in generalised inverse limits over Hausdorff continua. This generalises a result by Greenwood and Kennedy for generalised inverse limits over intervals. Connectedness is a property that has attracted much interest in generalised inverse limits, as these are not necessarily connected, unlike the inverse limits of continuous functions, which are. In the final chapter we prove a result relating to path connectedness in generalised inverse limits. Path connectedness in generalised inverse limits is in some ways a very different property to connectedness. For example, a generalised inverse limit may not be path connected even though all its finite approximants are path connected. This cannot happen if we replace the words "path connected" with "connected" in the previous sentence. The result proved in this chapter links the path connectedness of a generalised inverse limit with path connected properties of its finite approximants.