Homework 7: Product Topology

Instructions. Assignments should be stapled and written clearly and legibly.

1. Section 3.2, #3.15, 3.16, 3.19, 3.21. Note the problems 3.15 and 3.19 require proofs.
2. Let \(X\) be a topological space. Define the diagonal of \(X\) to be \(\Delta=\{(x,x): x\in X\}\), a subset of \(X\times X\). Prove that \(X\) is Hausdorff if and only if \(\Delta\) is closed in \(X\times X\).