An Arc as the Inverse Limit of a Single Nowhere Strictly Monotone Bonding Map on [0, 1]

Introduction. It is well known that a nondegenerate continuum (compact, connected metric space) is chainable if and only if it is homeomorphic to the limit of an inverse sequence of arcs with bonding maps onto. Several authors have studied chainable continua as inverse limits of arcs and in particular Henderson [3] and Mahavier [4] have contributed to the problem of characterizing the class C of chainable continua that are homeomorphic to the inverse limit of arcs using a single bonding map. Henderson showed that the pseudoarc is a member of C and Mahavier gave some examples of chainable continua not in C but proved that each chainable continuum is embedded in some member of C. In the inverse limit description of a chainable continuum the bonding maps mimic the pattern (see Bing [l ] for terminology related to chains) of the chains. Thus, one expects (proved by Capel [2]) an inverse limit with monotone bonding maps to be an arc where a monotone map corresponds to a chain going straight through another chain. Likewise the crooked chains that yield the pseudo-arc give rise to maps with corresponding "crooked" graphs. Thus, the question arises as to what kind of maps will yield an arc or more specifically what kind of single bonding map will yield an arc? In particular, is it possible for an arc to be the inverse limit of a nowhere strictly monotone map on [0, l]? As the above title indicates, the answer is yes. The author is grateful to the referee for suggestions regarding the proof of Theorem 1.