Neighboring mapping points theorem

Let f be a continuous map from a metric space X to M. We say that points in a subset N of X are f-neighbors if there exists a sphere S in M such that f(N) lies on S and there are no points of f(X) inside of S. We prove that if X is a unit sphere of any dimension and M is a contractible metric space then there are two f-neighbors in X such that the distance between them is greater than one. This theorem can be derived from the fact that for every non-null-homotopic closed covering C of X there is a set of f-neighbors N in X such that every member of C contains a point of N.Comment: 7 pages, 1 figur