  We examine properties of equidistant sets determined by nonempty disjoint
compact subsets of a compact 2-dimensional Alexandrov space (of curvature
bounded below). The work here generalizes many of the known results for
equidistant sets determined by two distinct points on a compact Riemannian
2-manifold. Notably, we find that the equidistant set is always a finite
simplicial 1-complex. These results are applied to answer an open question
concerning the Hausdorff dimension of equidistant sets in the Euclidean plane.
