Reduction of Open Mappings

In 1936 G. T. Whyburn [8] defined an irreducible mapping to be a mapping/, of a compact metric continuum A onto B, such that no proper sub-continuum of A maps onto B under/. Similarly, assuming A to be only compact and metric, he defined / to be strongly irreducible provided that no closed proper subset of A maps onto B. J. Rozanska [7] independently defined this second type of mapping in 1937. However, neither of these early papers studied these concepts per se, except that Whyburn pointed out that the Brouwer Reduction Theorem does guarantee that either of these two phenomena can be obtained by perhaps reducing A to a subset, A'. In 1939 Whyburn published a paper [9] specifically about these mappings and proved that for / to be strongly irreducible it is necessary and sufficient that the set D, of all points at which/ is one-toone, be dense in X. Also proved were certain other characterizations and another theorem which is of special interest here. This theorem fits into a sequence of theorems beginning with a question raised by W. A. Wilson [11 ] in 1935. As a setting for these theorems, suppose that/ is an open mapping of a compact metric space X onto Y. Wilson's question can be asked as follows: if Fis an arc, can/ be monotone and irreducible though not a homeomorphism? B. Knaster [S], in 1935, published an example where this is indeed the case. However, in this example the inverses of points of Y are mixed, some being arcs and others indecomposable continua. Knaster then posed the question as to whether there exists an example in which all inverses are homeomorphic, and in particular, are all arcs. The question concerning arcs as inverse sets was answered in the negative by E. E. Moise [6] in 1949 and the more general question is answered in the affirmative (as Eldon Dyer has pointed out) by an example that R. D. Anderson [l ] has announced in another connection, in which the inverses of points are pseudo-arcs. In the meantime certain gaps between these two results have been filled in by authors who were concerned with slightly different hypotheses on X, Y, and /. In 1939 Whyburn [9] proved that if Y is unrestricted,/ is light,