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.