Positive integers n which allow non-zero-dimensional, arc-free rim-n separable metric spaces

Common rim-finite spaces seem to always contain arcs and, in fact, Ward showed [4] that any non-trivial rim-finite continuum must contain an arc. However, the compactness condition cannot be removed completely. In [3] a non-compact non-zero-dimensional, rim-64, arc-free separable metric space is given. Hence there are natural numbers n for which rim-n, non-zero-dimensional, arc-free separable metric spaces exist. In this paper, we determine precisely which values of n allow this. For n≥3 there are separable metric, non-zero-dimensional arc-free spaces which are rim-n (and not rim-(n−1)). We also prove that any non-zero-dimensional and rim-2 space must contain an arc, and any rim-1 space must necessarily be zero-dimensional.