## Assignments

Assignments are due at the

start of class on the due date. (Plan ahead!)
Please staple and be organized.

One, due 23 September:
- From the textbook: 2D; 2G(1-3);
3A(1).

- The Zariski Topology: Let \(F\) be a field and \(S\subseteq
F[x_1,\ldots,x_n]\) for some \(n\geq1\). Show that the sets
\[V(S)=\left\{x\in F^n\colon f(x)=0\text{ for all }f\in S\right\}\]
form the closed sets of a topology on \(F^n\).

Two, due 7 October:
- From the textbook: 4G(1,2), 5D, 7J.

Three, due 16 October:

- Show that a function \(f\colon X\to Y\) is continuous if and only
if \(f^{-1}(U)\) is open in \(X\) for each open set \(U\) from a basis
for \(Y\).
- Show that, for any group \(G\), the profinite topology makes
\(G\) a topological group. Solution.
- From the textbook: 9F.

Four,
due 4 November:

- From the textbook: 10B, 11A(1), 13C(1,2).
- Show that, if \(G\) is a topological group which is \(T_0\), then
it is also \(T_2\). Solution.

Reading assignment for Math 9021 students: Profinite
groups
Five, due 20 November:
- From the textbook: 17A(1,2), 17N
- Show that if \(Y\) is compact, then for any \(X\), the projection
\(\pi_1\colon X\times Y\to X\) is a closed map.
- Recall that \(O(n)\) is, by definition, the set of \(n\times n\)
orthogonal matrices with real entries. The usual topology on \(O(n)\) is
obtained by noticing that it is a subspace of Euclidean space. (This
also makes it a topological subgroup of \(GL_n(\mathbf R)\).)

Prove that \(O(n)\) is compact.

Six, for practice only:

- From the book: 26H, 27B, 32A.