Openness of induced mappings
The authors investigate, for a map \(f : X \to Y\) between continua, when the induced map \(C(f): C(X) \to C(Y)\) is open. If \(X\) is either hereditarily locally connected or a fan then \(C(f)\) is open only when \(f\) is a homeomorphism. If \(X\) and \(Y\) are nondegenerate continua and \(f: X \times Y \to X\) is the natural projection, then \(C(f)\) is open only if \(X\) is hereditarily unicoherent. An example is constructed to show that hereditary unicoherence of \(X\) is not a sufficient condition for \(C(f)\) to be open. It is shown that for any dendrite \(X\) and nondegenerate continuum \(Y\) the corresponding map \(C(f)\) is open. Examples are also constructed involving continuous decompositions rather than products. Several examples and open questions are presented.
scientific article; zbMATH DE number 1367556