Download Practice Exam 5: Topology

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.
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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.
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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
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