Positive integers \(n\) which allow non-zero-dimensional, arc-free rim-\(n\) separable metric spaces (Q2344849)

For any natural number \(n\), a topological space is \textit{rim-\(n\)} if it possesses a base of open sets with boundaries of cardinality \(\leq n\); the space is \textit{rim-finite} if it is rim-\(n\) for some \(n\). An \textit{arc} is any Hausdorff continuum with exactly two non-cut points; the authors address the question of when a rim-finite space contains an arc. It is known [\textit{L. E. Ward jun.}, General Topology Appl. 6, 183--190 (1976; Zbl 0323.54006)] that every nondegenerate rim-finite Hausdorff continuum contains an arc, and it seems reasonable to expect the same for lots of other rim-finite spaces. This expectation is largely dispelled with the following results: (1) For \(n\geq 3\) there are separable metric spaces that are rim-\(n\), arc-free, and not rim-\((n-1)\). (2) Any rim-2 separable metric space must either contain an arc or be zero-dimensional (i.e., rim-0). (3) Any rim-1 Hausdorff space is rim-0, and therefore arc-free.