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.