Separating subsets and stable values (Q1868028)

Let \(Y\) be a metric continuum. A subcontinuum \(A\) of \(Y\) is said to be terminal provided that any subcontinuum of \(Y\) that intersects both \(A\) and \(Y\setminus A\) contains \(A\). A stable value of a mapping \(f: X\to Y\) is a point \(p\in f(X)\) such that there exists \(\varepsilon> 0\) with the property that \(p\) is a value of all mappings within a distance of \(\varepsilon\) of \(f\) in the supremum metric. It is shown that if a continuum \(Y\) contains pairwise disjoint terminal subcontinua \(A_1,\dots, A_n\) such that \(Y\setminus(A_1\cup\cdots\cup A_n)\) is disconnected, then each mapping \(f: X\to Y\) from a metric continuum \(X\) onto \(Y\) has a stable value. This generalizes a result of \textit{S. B. Nadler, jun.} [ibid. 126, 429-444 (2002; see the preceding review Zbl 1020.54022)].