On piecewise confluent mappings (Q2714186)

The authors consider the following problem of W. T. Ingram, which is problem 35 from the Houston Problem Book, and some variations of the problem [\textit{H. Cook, W. T. Ingram} and \textit{A. Lelek}, Lect. Notes Pure Appl. Math. 170, 365-398 (1995; Zbl 0828.54001)]. ``Suppose \(f\) is a continuous mapping of a continuum \(X\) onto a continuum \(Y\), \(Y=H\cup K\) is a decomposition of \(Y\) into subcontinua \(H\) and \(K\), \(f\mid f^{-1}(H)\) and \(f\mid f^{-1}(K)\) are confluent, and \(H\cap K\) is a continuum which does not cut \(Y\) and is an end continuum of both \(H\) and \(K\). Is \(f\) confluent?''NEWLINENEWLINENEWLINEThe authors show if one uses one definition of end continuum the conditions in the problem cannot be satisfied. However, if one uses the following definition of end continuum, then the answer to the problem is ``yes''. A subcontinuum \(P\) of a continuum \(X\) is said to be an end continuum in \(X\) provided that for every two subcontinua \(K\) and \(L\) of \(X\) the condition \(P\subset K\cap L\) implies that either \(K\subset L\) or \(L\subset K\).