Rearrangement of vector series. I

Let R-*(d) = R-d boolean OR (*) be the one-point compactification of Euclidean space R-d and d greater than or equal to 2. Given a permutation f of the set N of positive integers, let C-f(R-*(d)) denote the set of all sets C subset of or equal to R-*(d) for which there is a series Sigmaa(n) in R-d with zero sum such that C is the cluster set in R-*(d) of the sequence of partial sums of Sigmaa(f(n)). Every C is an element of C-f(R-*(d)) is non-empty connected and closed in R-*(d). We give a combinatorial characterization of the permutations f for which all non-empty closed connected subsets of R-*(d) belong to C-f (R-*(d)). For every permutation f of N, we determine all C is an element of C-f(R-*(d)) which contain *.