Exercise 2, § 4

A point $x$ in an irreducible space X is called generic if $\{x\}$ is everywhere dense in $X$.

(a) If X is a Kolmogoroff space (General Topology, Chapter I, § 1, Exercise 2), it admits at least one generic point; if X is an accessible space (General

Topology, Chapter I, § 8, Exercise 1), it cannot admit a generic point unless it consists of a single point.

(b) Give an example of an infinite irreducible accessible space (cf. General Topology, Chapter I, § 8, Exercise 5).

(c) Let X, Y be two irreducible spaces each admitting a generic point; suppose also that Y admits only one generic point y. Let $f : X \to Y$ be a continuous mapping; for $f(X)$ to be dense in Y, it is necessary and sufficient that, for every generic point x of X, $f(x) = y$.

(d) Let $(X_\alpha, f_{\alpha\beta})$ be an inverse system of irreducible spaces whose indexing set is directed; suppose that each of the $X_\alpha$ admits a single generic point $x_\alpha$. Show that, if, for $\alpha \leq \beta, f_{\alpha\beta}(X_\beta)$ is dense in $X_\alpha$, then $X = \lim \leftarrow X_\alpha$ is irreducible and admits a single generic point (same method as in Exercise 1).