Non-degenerate Maps and Sets

We construct certain non-degenerate maps and sets, mainly in the complex-analytic category. For example, we show that for every countable subset S in an irreducible complex space X there exists a holomorphic map from the unit disk to X such that S is contained in the image. Furthermore, given an irreducible complex space X, there is always an infinite subset S such that for every proper analytic subspace Z of X the intersection of S with Z is finite.