Texas A&M University Department of Mathematics Volodymyr Nekrashevych Fall 2015 Math 636 — Problem Set 10 Issued: 11.29 Due: 12.15 10.1. Let X be a set. Describe the smallest T1 topology on X. 10.2. Let f and g be continuous maps from a topological space X to a Hausdorff space Y . Prove that if f (x) = g(x) for all x ∈ A, where A ⊂ X is dense, then f (x) = g(x) for all x ∈ X. 10.3. Let X be a Hausdorff space such that every proper closed subset of X is compact. Prove that X is compact. 10.4. Let {Y Qi }i∈I be a collection Qof topological spaces and let Ai ⊂ Yi . Prove that i∈I Ai is dense in i∈I Yi if and only if Ai is dense in Yi for all i ∈ I. 10.5. Suppose that M and N are smooth manifolds with M connected, and F : M −→ N is a smooth map such that F∗ : Tp M −→ TF (p) N is the zero map for each p ∈ M . Show that F is a constant map. 10.6. Let F : R2 −→ R be defined by F (x, y) = x3 + xy + y 3 + 1. For which t the sets F −1 (t) are embedded submanifolds of R2 ? 10.7. Show that there is a smooth vector field on S 2 that vanishes at exactly one point. 10.8. If C is a circle embedded smoothly in R4 , show that there exists a three-dimensional hyperplane H such that the orthogonal projection of C onto H is an embedding.