I am trying to relate two (maybe not) different decompositions of a noetherian topological space into irreducible subsets, given in Ravi Vakil's notes on algebraic geometry.
Exercise 4.6.N : Let $X$ be a topological space, then any point is contained in a an irreducible component.
It follows that any space $X$ is the union of its (closed) irreducible components, but there is not statement of uniqueness.
Proposition 4.16.14 : Let $X$ be a noetherian topological space, and $Z \subseteq X$ a closed (non-empty) subset. Then there is a unique decomposition $Z = Z_1 \cup \cdots \cup Z_n$ where the $Z_i$'s are irreducible closed subsets, none containing another.
So in particular $X$ is a finite union of irreducible closed sets, but here it seems like they may not be its components.
The exercise is just a small exercise, while the proposition seems to be more important. So my question is : Why is the decomposition in the proposition more important than the one in the exercise ?
Is it because of the uniqueness statement ? But I think the decomposition of $X$ into its irreducible closed components also satisfies this uniqueness condition.
Is it because it applies to any closed subset $Z$ ? But $Z$ also has the decomposition of the exercise. However, the subsets are irreducible in $Z$ and not in $X$, so this is the point ?
Thank you for your help.