Introduction. Let n be a positive integer, and let yn be a topological space with the following property: a topological space X has dimension Yn can be extended over X. Then we call yn a test space for dimension n. In a previous paper [10], we characterized test spaces for dimension n under the assumption that both X and yn were separable metric. In addition, we required Yn to be compact, along with some other side conditions. In the present paper, we are able to improve the characterization in two ways. First, we remove the separability requirement from both X and Yn, and secondly, the compactness requirement on yn is removed. The resulting space Yn is called a test space for metric spaces of dimension n, or simply a test space for metric spaces. The main reason that we required yn to be compact in [10] was that compactness was needed to apply S. T. Hu's theorem on the homotopy type of spaces [8]. In the present paper we employ a theorem of J. H. C. Whitehead [11, Theorem 3, p. 216] which allows us to lift this restriction.