The impossibility of filling $E^n$ with arcs
Stephen L. Jones · Bulletin of the American Mathematical Society · 1968
The purpose of this paper is to outline a proof of the following MAIN THEOREM.Iff is a closed continuous map ofE n onto any space S, then some point in S has an inverse image which is not an arc.In 1936 J. H. Roberts [l] showed that there does not exist an upper semicontinuous (use) collection of arcs filling the plane.Recently L. B. Treybig [2] has obtained some partial results for polygonal arcs in £ w .In 1955 Eldon Dyer [3] outlined a proof that there is no continuous decomposition of E n into arcs.This proof incorporates some of the ideas of both Roberts and Dyer.We will suppose that all statements are for E n for a given n.DEFINITIONS.If C/and F are sets with disjoint closures, we say that an arc a has k folds between U and V if a contains k + l disjoint subarcs between U and V. Furthermore, if the distance between each pair of the k + l subarcs is greater than e, we say that the width of the folds is greater than e.If a contains a subarc which has endpoints in U and which intersects V, then a is said to have a fold with the bend in V.If K is a set, €>0, let N € (K) denote the open e-neighborhood of K in E n .If H is a collection of sets, let H* denote the set of all points covered by elements of H.Suppose A is compact and B is a closed subset of A. If any two points of E n -A which are separated by A are also separated by B, then B is said to be essential in A. If H is a use collection of arcs and points filling A and B intersects each element of H, then B is said to be full in A H .If B meets each element of H in a continuum, then B is said to be a quasi-section of A H .Assume H is a use collection of arcs and points filling the compact set-ST.The proof is an exercise in the Vietoris mapping theorem on the Cech homologies of -X", F, and the decomposition space.LEMMA 2. If K is full in X H , U is open f TJC\K~0, and no element