Continuity of Jones' function is not preserved under monotone mappings (Q2411461)

In the present paper the authors study Jones' function, which is defined in the following way: for a continuum \(X\) and \(A \subset X\) set \[ T(A) = \{p \in X \;: \;\text{for each } M \in C(X) \text{ with } p \in \text{int}_X(M), M\cap A \neq \emptyset \} \] (note that Jones' function was introduced in [\textit{F. B. Jones}, Am. J. Math. 70, 403--413 (1948; Zbl 0035.10904)]). Since there are only few families of continua for which it is known that \(T\) is continuous, the authors observe properties of \(T\) and of a continuum \(X\) under the assumption that \(T\) is continuous for \(X\) (Proposition 2.1 and Theorems 2.2--2.4). The main result of the paper can be found in the third section. It is a construction of a continuum \(X\) for which \(T_X\) is a continuous function such that \(X\) contains an arc \(L\) with the property that if \(Z=X/L\) is the space obtained by shrinking \(L\) to a point, then \(T_Z\) is not continuous for \(Z\). The construction of \(X\) is highly non-trivial, but the authors describe all technical details very carefully. This example also answers Bellamy's question (Problem 155 in [``Continuum Theory Problems'', Topol. Proc. 8, No. 2, 361--394 (1983; Zbl 0544.54001)], also stated as Problem 1.4 in the paper) in the negative.