Induced MO-mappings (Q1961894)

All spaces are metrizable. The authors study the general question how properties of induced hyperspace maps can be translated in terms of properties of the original map, and vica versa. A continuous function \(f: X\to Y\) is said to be an MO-mapping provided that it can be written in the form \(f= f_1\circ f_2\) such that \(f_1\) is open and \(f_2\) is monotone. Similarly for OM. It is known that if \(f: X\to Y\) is surjective and continuous, then \(f\) is OM iff \(2^f\) is OM iff \(C(f)\) is OM. For MO the situation is different. The authors prove that if \(f\) is MO then so is \(2^f\). The converse is an open problem. For \(C(f)\) a similar result does not hold. They present examples of open maps \(f\) from \([0,1]\) onto itself such that \(C(f)\) is not MO.