Exercise 3, § 4

Let X be a topological space and $(Y_\alpha)$ a right directed family of subspaces of X; show that if each of the subspaces Y, is irreducible and X is the union of the family $(Y_\alpha)$, X is irreducible. Deduce an example where X is a Kolmogoroff space, each of the Y, admits a generic point, but X does not admit a generic point.