# I.1.10

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This problem has Hartshorne Height 2; problem I.1.6 can help solve this one. No big commutative algebra results to apply, just work through the topology.

### HAPPY

a) Use exercise I.1.6 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.