Definition. If $X$ is a topological space, we define the dimension of $X$ (denoted $\operatorname{dim} X$ ) to be the supremum of all integers $n$ such that there exists a chain $Z_{0} \subset Z_{1} \subset \ldots \subset Z_{n}$ of distinct irreducible closed subsets of $X.$
now let For $n \in \mathbb{Z}_{\geq 0},$ let $U_{n}=\{n, n+1, n+2, \ldots\} .$ Then the set $\tau=$ $\left\{\emptyset, U_{0}, U_{1}, \ldots\right\}$ is a topology of open sets on $\mathbb{Z}_{\geq 0}$. In this space, if $C$ and $C^{\prime}$ are closed sets, then it is easy to see that either $C \subset C^{\prime}$ or $C^{\prime} \subseteq C,$ that every nonempty closed set is irreducible, and that every closed set other then $\mathbb{Z}_{\geq 0}$ is finite. So this is an example of a Noetherian infinite dimensional topological space.
how we can see that either $C \subset C^{\prime}$ or $C^{\prime} \subseteq C,$ that every nonempty closed set ? why this topology is Noetherian ? why this topological space is infinite dimensional ?
As a hint, the closed set $C_n$ corresponding to $U_n$ is $\{0,\ldots,n-1\}$ und the sets $U_n$ form a basis of the topology.