Exercise 4, § 4

Let Y be an irreducible space admitting a single generic pointy, X a topological space and $f : X \to Y$ a continuous mapping.

(a) For every irreducible component Z of X meeting $\overline{f^{-1}(y)}$, $f(Z)$ is dense in Y.

(b) Give an example where $f(X)$ is dense in Y but $\overline{f^{-1}(y)}$ is empty (take X to be a subspace of Y).

(c) Show that, if Z admits a generic point z and $f(Z)$ is dense in Y, then $f(z) = y$ and z is a generic point of $Z \cap \overline{f^{-1}(y)}$.