The Stacks project

Example 5.11.3. Let $X = [0, 1]$ be the unit interval with the following topology: The sets $[0, 1]$, $(1 - 1/n, 1]$ for $n \in \mathbf{N}$, and $\emptyset $ are open. So the closed sets are $\emptyset $, $\{ 0\} $, $[0, 1 - 1/n]$ for $n > 1$ and $[0, 1]$. This is clearly a Noetherian topological space. But the irreducible closed subset $Y = \{ 0\} $ has infinite codimension $\text{codim}(Y, X) = \infty $. To see this we just remark that all the closed sets $[0, 1 - 1/n]$ are irreducible.