Introduction and preliminaries. Generalizing the CEapproximation theorem of Arment rout [1, 2] and Siebenmann [20] for finite-dimensional manifolds, Ferry proved an a-approximation theorem for Q-manifolds in [8] and an «-approximation theorem for manifolds of dimensions > 5 in a joint work with Chapman [6]. Recently, the author proved in [16] an a-approximation theorem for Q°°-manifolds: "Given an open cover a of a Q°°-manifold N, then there is an open cover ß of N such that every ^-equivalence

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... from a Q°°-manifold M to N is a-close to a homeomorphism". It will be shown in this note that such an ^-approximation theorem also holds true for R°°-manifolds. So, the question (NLC 8) in [9] has an affirmative answer. As in [16] , in the process of proving the main theorem, some results similar to a few properties of Z-sets in Q and l{R n }