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Abstract. Let f : X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g :These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij about extensional dimension is also established. Introduction.All spaces are assumed to be completely regular and all maps continuous. This paper is concerned with the following two results. The first one was proved by Pasynkov [25] (see [24] for noncompact versions) and the second one by Toruńczyk [31]:

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Here are two of our main results: Theorem 1. Let X be a normal space with dim X = n and m ≥ n + 1.Then the space C * (X, R m ) of all bounded maps from X into R m equipped with the uniform convergence topology contains a dense G δ -subset consisting of maps g such that g(X) ∩ Π d is at most (n + d − m)-dimensional for everyTheorem 2. Let X be a metrizable compactum with dim X ≤ n and m ≥ n + 1. Then, C(X, R m ) contains a dense G δ -subset of maps g such thatIn case m = 2n + 1, the combination of Theorem 1 and the Nöbeling-Pontryagin embedding theorem provides a generalization of a theorem due to Roberts [20]. Theorem 2 extends the following results: the Nöbeling-Pontryagin embedding theorem (d = 0, m = T ≥ 2n + 1); Hurewicz's theorem [15] about mappings into an Euclidean space with preimages of small cardinality (d = 0, n + 1 ≤ m = T ≤ 2n); Boltyanski's theorem [6, Theorem 1] about k-regular maps (d = k − 1, t = 0, T = m ≥ nk + n + k) and Goodsell's theorem [12] about existence of special embeddings (t = 0, T = m). An infinite-dimensional analogue of Theorem 2 is also established. Our results are based on Theorem 1.1 below which is considered as a converse assertion of the transversal Tverberg's theorem and implies the Berkowitz-Roy theorem [1], [12].

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