MONOTONE MAPS AND E-MAPS

In this paper, a compactum is a compact metric space; a continuum is a connected compactum, and a map is a con tinuous function. A map f: X + Y is an £-map, for £> 0, if and only if diam(f-l(y » < £ for each y E Y. If X is a continuum, and ~ is a class of continua, then X is ~-like if and only if there is an £-map from X onto some element of ~ for every £> O. If ~ = {y} is degenerate, then X is also said to be Y- like. A graph is a sp~~,.e homeomorphic to the space of a one-dimensional finite complex.. It will usually be assumed that the graph is triangulated, and the I-simplexes will also be called edges and the O-simplexes, vertices. In [2], Carlisle proved a theorem that related the problem of determining which graphs are "like! " which other graphs to the intuitively simpler problem of determinin