I.1.10

This problem has Hartshorne Height 1. No big commutative algebra results to apply, just work through the topology.

Tips
a) Use exercise I.1.7 and show if $$Z \subset Z' $$ are distinct and closed in an arbitrary subset $$Y \subset X$$ then $$ \overline{Z}, \ \overline{Z'}$$ are distinct in X.  In fact take a point of $$Z'$$ that is not in $$Z$$ and show this point is still not in the closure.

b) The hard part is to show there is an i such that $$\dim X \le \dim U_i$$. The idea is to take a chain or closed irreducible subsets in X, $$Z_0 \subset ... \subset Z_n $$ and pick a $$U_i$$ intersecting $$Z_0$$.  Now show the chain still has the same length as a chain in $$U_i$$.  Basically if this wasn't the case you can find two disjoint open subsets in an irreducible closed subset.

c) It can be done by constructing a simple two point topological space.

d) If $$Y \ne X$$ then for every chain in Y, there is a strictly longer chain in X by taking X to be the last element of the chain.

e) Think: countably many finite dimensional noetherian topological spaces.