Exercise 12, § 2

Let $f : X \to Y$ be a surjective continuous mapping. Prove that the following two properties are equivalent: $\alpha)$ for every closed subset $F \neq X$ of $X$, one has $f(F) \neq Y$; $\beta)$ for every open subset $G$ of $X$, one has $f(G) \subset Y - f(X - G)$. A continuous mapping with this property is called irreducible.