## 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.