General Topology/Miscellaneous spaces

Noetherian spaces
In the latter case, we say that the ascending chain $$U_1 \subseteq U_2 \subseteq \cdots \subseteq U_n \subseteq \cdots$$ stabilizes.

Irreducible spaces
That is, every open set contains every generic point.

Exercises

 * 1) Suppose that $$\mathbb N$$ is equipped with the cofinite topology. Prove that this topological space is irreducible, but does not admit a generic point.
 * 2) Prove that on the two-point space $$X = \{0,1\}$$ one may find a topology that makes $$X$$ into an irreducible space with two generic points. Generalize this example to any set.