Maps between continua with stable values
Let \(X\) and \(Y\) be metric continua. 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. If \(f\) is onto, then \(f\) is called a pinch at \(p\in Y\) provided that there exist a continuum \(M\) and onto mappings \(g: X\to M\) and \(\pi: M\to Y\) such that \(f= \pi\circ g\) and there are points \(q_1,q_2\in M\) with \(\pi(q_1)= \pi(q_2)\) and \(\pi|_{(M\setminus\{q_1, q_2\})}\) is a homeomorphism onto \(Y\setminus\{p\}\). It is shown that if \(X\) and \(Y\) are Peano continua and a local separating point \(p\in Y\) is an instable value for some onto mapping \(f: X\to Y\), then \(f\) is a pinch at \(p\). Consequently, each local separating point is a stable value for any monotone mapping between Peano continua. The main result says that if \(X\) and \(Y\) are Peano continua, \(F\) is a finite separator of \(Y\) and \(f: X\to Y\) is onto, then some point of \(F\) is a stable value of \(f\). This implies that any mapping of a Peano continuum onto a regular curve has a stable value. In the last section continua are investigated that belong to Class(S), that is, such continua \(Y\) that every mapping of any continuum onto \(Y\) has a stable value. Among other results it is shown that every hereditarily decomposable arc-like continuum is in Class(S).
scientific article; zbMATH DE number 1900964
scientific article