Certain collections of arcs in 𝐸³

Abstract

t. Introduction. In considering upper semicontinuous decompositions of E3, it is sometimes useful to know whether a given collection of continua can be transformed, by a homeomorphism of E3 onto itself, into another collection which is simpler in some respects; for example, a collection of straight line intervals might be transformed into a collection of vertical intervals, or a collection of arcs into a collectioin of straight line intervals. It might also be useful to know conditions under which such a transformation can be effected by means of a particular type of homeomorphism of E3 onto itself. In this paper, the following questions of this type will be considered. Suppose a and ,3 are horizontal planes and G is a continuous collection of mutually exclusive arcs, each of which is irreducible from a to / and no one of which contains two points of any horizontal plane, such that the sum of the elements of G is compact and intersects a in a totally disconnected set. Under what conditions is there a homeomorphism of E3 onto itself which takes each element of G onto a vertical interval and does not change the z-coordinate of any point? It is shown, with the aid of certain results due to Bing [1] and Fort [5 ], that such a transformation is not always possible, even when the elements of G are straight line intervals. The following condition is found to be necessary and sufficient for the existence of such a transformation (see ?3 for definitions of unfamiliar terms): For every positive number e there exists a finite set K1, K2, * .* Kn of topological cylinders with bases on a and : such that (1) the solid cylinders determined by K1, K2, , Kn. are mutually exclusive, (2) each arc of G is enclosed by some K? and (3) each Ki has horizontal diameter less than e.