Inverse limits of set-valued functions indexed by the integers

The notion of an inverse limit using set-valued functions was introduced by Mahavier in [3], and was further explored by Mahavier and Ingram in [2]. Illanes has shown that a circle cannot be expressed as an inverse limit of an arc indexed by the natural numbers using a single set-valued bonding map [1], and Nall has shown that we cannot use a similar construction to produce any finite graph (other than an arc) [8], or an n-cell for n > 1 [6]. In this paper, we will explore inverse limits using the integers as an index set. We will present an example of a 2-cell obtained under this index. We will then show that the only finite graph that can be obtained as an inverse limit of arcs using this index is again an arc.