Constructing near-embeddings of codimension one manifolds with countable dense singular sets

We present, for all n ≥ 3, very simple examples of continuous maps f : Mn-1 → Mn from closed (n-1)-manifolds Mn-1 into closed n-manifolds Mn such that even though the singular set S(f) of f is countable and dense, the map f can nevertheless be approximated by an embedding, i.e. f is a near-embedding. In dimension 3 one can get even a piecewise-linear approximation by an embedding