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.
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