Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Brouwer fixed-point theorem wikipedia , lookup
Continuous function wikipedia , lookup
Euclidean space wikipedia , lookup
Surface (topology) wikipedia , lookup
Orientability wikipedia , lookup
Metric tensor wikipedia , lookup
Fundamental group wikipedia , lookup
Geometrization conjecture wikipedia , lookup
Grothendieck topology wikipedia , lookup
Practice Exam 5: Topology June 3, 2005 Directions: This is a three hour closed book exam. There are two parts of the exam. Work four questions from part 1 and four questions from part 2. 1 Part 1: General Topology 1. Let X and Y be topological spaces and let f : X → Y be a function. Prove that the following are equivalent. • For all U ⊂ Y open, f −1 (U ) is open in X. • For all F ⊂ Y closed, f −1 (F ) is closed in X. • For all A ⊂ X, f (A) ⊂ f (A). 2. Prove that X is normal if and only for every closed set F and open set U with F ⊂ U , there exists V open with F ⊂ V ⊂ V ⊂ U . 3. Prove that every compact Hausdorff space is regular. 4. Give an example of a topological space that is connected but not path connected. Justify your answer. 5. Suppose that (X, d) is a metric space and A ⊂ X. Prove that the point p is in A if and only if there is a sequence of points an ∈ A with limn→∞ an = p. 6. Prove that the space X is Hausdorff if and only if the diagonal of X ×X is closed. 1 2 Part 2: Smooth Manifolds 1. Prove that if F : M → N is a smooth map of smooth manifolds and Q is a regular value of F then F −1 (Q) is a smooth submanifold of M . 2. Let V and W be vector spaces and A : V → W be a linear map. Prove that if ω, η ∈ Λ∗ (W ) then A∗ (ω ∧ η) = A∗ (ω) ∧ A∗ (η). 3. Produce a C ∞ -compatible atlas of coordinate charts for S n , the set of vectors of length one in Rn+1 to show that it is a smooth n-manifold. Justify your answer. 4. Does there exist a submersion f : T 2 → R2 ? By T 2 we mean S 1 × S 1 given the smooth structure as a cartesian product of copies of S 1 . Justify your answer. 5. State and derive the transformation law for covectors. 6. Let ω = zdx ∧ dy − xdx ∧ dz + xdy ∧ dz be a two form in the standard (x, y, z) coordinates on R3 . Let σ : [0, 1]2 → R3 be the singular two cube, σ(s, t) = (s, t, s2 + t2 ). Compute Z ω. σ 2