Instructions: Do three problems from Part I, and three problems from Part II. If you do more than three problems from either section, you must indicate which three you want to count.

## Part I: Topology (970)

1. (a) Suppose that $( X , \tau )$ and $( Y , T ^ { \prime } )$ are topological spaces, $A \subseteq X$ , and $f : ( X , T ) \to ( Y , T ^ { \prime } )$ is a homeomorphism. Show that $f | _ { A } : ( A , { \mathcal { T } } _ { A } ) \to ( f ( A ) , { \mathcal { T } } _ { f ( A ) } ^ { \prime } )$ is also a homeomorphism (using the corresponding subspace topologies).

(b) Show that there is no homeomorphism $f : [ 0 , 1 ) \to ( 0 , 1 )$ (where each has the subspace topology it inherits from the usual topology on R).

2. Show that if the metric space $( X , d )$ is separable (i.e., it contains a countable dense subset), then the metric topology on X is second countable.

3. Let $( X , \tau )$ be a Hausdorff space and let ${ \mathcal { T } } ^ { \prime } = \{ { \mathcal { U } } \subseteq X : X \setminus { \mathcal { U } } \subseteq X $ is $\operatorname { c o m p a c t } \} \cup \{ \varnothing \}$ . Show that ${ \boldsymbol { \mathcal { T } ^ { \prime } } }$ is a topology on X, and is coarser than T . Show that, in general, they need not be equal.

4. Two subsets $A , B \subseteq X$ of the space $( X , \tau )$ are called separated if there are $U , V \in \mathcal { T }$ with $A \subseteq U \subseteq$ $X \setminus B$ and $B \subseteq V \subseteq X \setminus A$ . We say X is completely normal if X is $T _ { 1 }$ and if for every pair of separated subsets $A , B$ , there are $U , V \in \mathcal { T }$ so that $A \subseteq U , B \subseteq V$ , and $U \cap V = \emptyset$ . Show that a space $( X , \tau )$ is completely normal $\Longleftrightarrow \mathrm { e v e r y }$ subset of X is normal.

5. Show that if $f : ( X , T )  ( Y , T ^ { \prime } )$ is a homotopy equivalence, then f induces a bijective correspondence between the path-components of X and the path-components of Y .

## Part II: Algebraic Geometry (953)

1. Prove that the ideal $( x ^ { n } + y ^ { n } + z ^ { n } ) \subset \mathbf { C } [ x , y , z ]$ is a prime ideal for every integer $n > 0$

2. Find the multiplicity of each singular point of the zero locus of $x y z ( x + y ) - x ^ { 4 } - y ^ { 4 }$ in $\mathbf { C P ^ { 2 } }$

3. Let $C \subset \mathbf { C P } ^ { 2 }$ be the zero locus of a homogeneous polynomial of degree 5. Assume C is smooth. Let D be a divisor on $C$ of degree 11. What is the most you can say about dim Γ $\ " ( D ) ?$ (For example, is it positive? Can you give a lower bound? Can you give an upper bound? Can you determine it exactly?) Justify your answer.

4. Let $\mathcal { F }$ be the structure sheaf of $\mathbf { C P ^ { 2 } }$ , and let I be the sheaf of ideals defining two distinct reduced points $p , q$ of $\mathbf { C P ^ { 2 } }$ . Let $\mathcal { G }$ be the presheaf defined for any Zariski open set $\bar { U } \subset \mathbf { C P } ^ { 2 }$ as ${ \mathcal { G } } ( U ) =$ ${ \mathcal { F } } ( U ) / { \mathcal { T } } ( U )$

(a) Show that G is not a sheaf.

(b) Determine the associated sheaf $\mathcal { G } ^ { + } \mathrm { ~ ( i . e . }$ , for any Zariski open set $U \subset \mathbf { C P } ^ { 2 }$ , determine ${ \mathcal { G } } ^ { + } ( U ) )$

5. (a) Show that the closed points of any scheme of finite type over a field are dense.

(b) Give an example of a nonempty scheme whose closed points are not dense.