On Monotone Retractability into Simple Arcs

Radó has noted that (II) implies (r) and that the sphere and 2-cell each have (II). In this paper it will be shown that (1) for locally connected continua in general, property (II) is equivalent to unicoherence, (2) for plane locally connected continua, property (ir) is equivalent to unicoherence, and (3) every closed 2-dimensional connected manifold has property (x). To clarify our meaning, we recall that a continuum is compact, connected and metric. A continuous mapping f(A)~B on a continuum A is monotone provided f~(y) is a continuum for yÇzB. If