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In this paper we consider the relationship between the topological dimension ${\rm dim}_{\sf T}(X)$ and the lower and upper q-Rényi dimensions $\underline D_{q}(X)$ and $\overline D_{q}(X)$ of a Polish space X for q ∈ [1, ∞]. Let and denote the Hausdorff dimension and the packing dimension, respectively. We prove that $$\eqalign{{\rm dim}_{\sf H}(X) \le \underline{D}_{\infty}(X) \quad \le \underline{D}_{q}(X) \le \underline{D}_{1}(X),\cr \overline{D}_{\infty}(X)\quad \le \overline{D}_{q}(X) \le \overline{D}_{1}(X) \le {\rm dim}_{\sf P}(X),}$$ for all analytic metric spaces X (whose upper box dimension is finite) and all q ∈ (1, ∞); of course, trivially, $\underline D_{q}(X) \le \overline D_{q}(X)$ for all q ∈ [1, ∞]. As a corollary to this we obtain the following result relating the topological dimension and the lower and upper q-Rényi dimensions: $${\rm dim}_{\sf T}(X) = {\rm inf}_{X\sim Y} \underline {D}_{q}(Y), \quad (1)$$ $${\rm dim}_{\sf T}(X) = {\rm inf}_{X\sim Y} \overline {D}_{q}(Y), \quad (2)$$ for all Polish spaces X and all q ∈ [1, ∞]; in (1) and (2) we have used the following notation, namely, for two metric spaces X and Y, we write X ∼ Y if and only if X is homeomorphic to Y. Equality (1) has recently been proved for q = ∞ by Myjak et al.