Open maps of chainable continua

It is apparently “well known” that the image of the closed unit interval under an open map is homeomorphic to the closed unit interval (see [13], [11], and [15]). In this paper, we generalize this result to chainable continua. In particular, the fact that the open continuous image of a chainable continuum is also chainable is proved, answering a question of A. Lelek (see [10]). This fact, as well as its proof, implies that the open continuous image of the pseudo-arc is also a pseudo-arc. An additional corollary (of the proof) is that a local homeomorphism of a chainable continuum is actually a homeomorphism. The proofs are all very elementary.