Exercise 1, § 1

Let $B$ be a topological space, let $(X, p)$ and $(Y, q)$ be $B$-spaces and let $f: X\rightarrow Y$ be a mapping such that $q\circ f=p$. Let $A$ be a topological space and let $g: A\rightarrow B$ be a continuous mapping. Consider the $A$-morphism $f_A: X_A\rightarrow Y_A$ deduced from $f$ by change of base.

(a) Give an example where the mapping $f_A$ is continuous, the mapping $g$ is surjective but where the mapping $f$ is not continuous.

(b) Suppose that $g$ is surjective and universally strict and that $f_A$ is continuous. Prove that $f$ is continuous.

(c) Suppose that the mapping $g$ is surjective and that the mappings $q$ and $f_A$ are open. Prove that $f$ is open.