Characterizations of the continuous images of the pseudo-arc

Introduction.At the Summer Institute on Set Theoretic Topology, Madison, Wisconsin, 1955, the question "What characterizes all continuous images of the pseudo-arc [1; 7; 10]?" was raised by R. H. Bing [3].In particular it was asked whether or not there exists a characterization of the continuous images of the pseudo-arc analogous to the well-known Hahn-Mazurkiewicz characterization of the continuous images of the arc [5; 8].The purpose of this paper is to investigate this question and to initiate an investigation of the properties of the class of continuous images of all chainable continua in general.A related question considered is whether or not every chainable continuum is a continuous image of the pseudo-arc.In §3 of this paper two global characterizations of the continuous images of the pseudo-arc are establishedi1), and it is shown that if the definition of local connectedness is suitably reformulated, then these characterizations are analogous to that of Hahn and Mazurkiewicz for the arc.Furthermore, in §4, these characterizations prove to be readily applicable to showing that the pseudo-arc can be mapped continuously onto any chainable continuum.A consequence of this latter result is that the class of continuous images of the pseudo-arc and the class of continuous images of all chainable continua are identical.Next we present one of the most important results of this paper with respect to answering the question of whether or not there exists a characterization of the continuous images of the pseudo-arc analogous to that of Hahn and Mazurkiewicz for the arc.Specifically, in §5, we show that there does not exist any local topological property which characterizes all continuous images of the pseudo-arc.This simultaneously indicates that the original question of R. H. Bing has an essentially negative answer and that, in this respect, the characterizations of §3 cannot be improved.Finally, in §6, a series of examples is given to show that the Presented to the Society, January 29,1960 under the title Continuous mappings of the pseudoarc; received by the editors January 17, 1963.(!) The referee has pointed out that during the period of his examination of this paper an alternative development of one of these characterizations was given by A. Lelek, On weakly chainable continua, Fund.Math.51 (1963), 271-283.However, since the preliminary theorems of the present development as well as this characterization are necessary to the completion, in §5, of the answer to the question of R. H. Bing, this paper is presented essentially in its original form.