On inverse limit sequence

The following arose out of an unsuccessful attempt to answer the question "is there a map of the unit interval onto itself whose inverse limit is hereditarily indecomposable?" This question naturally leads to the broader problem of determining what sort of continua may be obtained by taking the inverse limit of a single map on the unit interval. A very limited number of answers to this problem will be found in Chapter IV, chiefly dealing with how to obtain indecomposable continua. Chapter V gives some examples to show why Chapter IV contains very little in the way of theorems characterizing the inverse limits by means of reasonable properties of the map. Some examples are also given of continua which may be obtained. A complete answer is given in Chapter III to the question of what may be obtained as the inverse limit of a sequence of functions on the unit interval. The answer is complete since it is that every compact chainable continuum may be so obtained, and only such continua may be obtained. The question of which compact chainable continuum one will get with a given sequence of maps is not answered. The study of inverse limits has developed in two principal directions. The first direction is abstract homology theory, which is the source of the concept. This direction will not be considered. The second direction is apparently an outgrowth of the first. It consists of giving examples of unusual continua conveniently generated as inverse limits and the study of the properties used in generating the examples.