Download Final Review

Final Review
The final exam will potentially cover everything we’ve gone over in class.
You should look over your notes, look over my notes, look over your old HW,
your midterms, and the solutions I posted.
Here are some sample problems.
Problem 1. Let B denote the collection of “open” equilateral triangles in R2
with one side parallel to the x-axis and pointing up (by open, I mean the triangle
without its boundary).
(a) Prove that B is a basis.
(b) Prove that B generates the standard topology on R2 .
Problem 2. Let X denote R2 with the dictionary order topology. Let A be
the subset given
A = (0, 1) × (0, 1) ⊂ R2 .
(a) Is A open?
(b) Is the point
1
2
× 0 in the closure Ā?
(c) Is the point 0 ×
1
2
in Ā?
(d) Describe Ā.
Problem 3. Suppose f : X → Y is continuous. Show that if x1 , x2 , . . . is a
sequence of points in X converging to x then the sequence f (x1 ), f (x2 ), . . . in
Y converges to f (x).
Problem 4. Suppose A ⊂ X and B ⊂ Y are closed subsets. Prove that A × B
is a closed subset of X × Y in the product topology.
Problem 5. Suppose p : X → Y is a surjective map from a topological space
X to a set Y . One can use this to define the quotient topology on Y . Prove that
the image of an open subset A ⊂ X under p is open in the quotient topology
on Y if A is symmetric with respect to p, which means that: if x ∈ A, then any
other y ∈ X with p(y) = p(x) is also in A.
Problem 6. Consider the equivalence relation on X = R in which any two
negative numbers are equivalent, any two positive numbers are equivalent, and
zero is equivalent to only itself. What is the set X ∗ of equivalence classes?
What are the open subsets of X ∗ in the quotient topology associated to the
map p : X → X ∗ which sends a number to its equivalence class?
Problem 7. Prove that Q ⊂ R is not connected.
Problem 8. Prove that if X is path connected then it is connected.
Problem 9. Prove that the image of a compact space under a continuous map
is compact.
Problem 10.
(a) Prove in the finite complement topology on R, every subspace is compact.
(b) Consider the function f : R → R given by f (x) = x, where the domain has
the finite complement topology and the range has the standard topology.
Is f continuous with respect to these topologies?
Problem 11. Let S 2 denote the unit sphere centered at the origin in R3 .
Suppose f : R3 → R is the function given by
f (x, y, z) = x2 + y 3 + z 5 .
Prove that there is some N ∈ R such that f (p) ≤ N for all points p ∈ S 2 .
Problem 12. Prove that the two subsets A, B ⊂ R2 in Problem 9 of Midterm
2 Review are not homeomorphic.
Problem 13. Suppose A is a star convex subset of Rn (as in Problem 1 of
Section 52). Prove that π1 (A, a) is trivial for any a ∈ A.
Problem 14. Suppose h : R2 → S 1 is a continuous map. Let f be a loop in
R2 based at the origin 0. Prove that h ◦ f is path homotopic to the constant
loop in S 1 based at f (0). (Hint: consider the induced map h∗ : π1 (R2 , 0) →
π1 (S 1 , f (0)).)
Problem 15. Prove that the map p : R → S 1 given by
p(t) = (cos 2πt, sin 2πt)
is a covering map.
Problem 16. Let p : R → S 1 be the covering map above, and consider the
induced covering map
q = p × p : R × R → S1 × S1.
(a) Find a lift starting at the origin in R2 of the loop f : I → S 1 × S 1 given
by
f (t) = (cos 2πt, sin 2πt) × (cos 4πt, sin 4πt).
Draw a picture of the image of this lift. Where does the lifting correspondence
φ : π1 (S 1 × S 1 ) → (p × p)−1 ((1, 0) × (1, 0))
(associated to lifts starting at the origin) send [f ]? Is [f ] the identity
element in the fundamental group of the torus S 1 × S 1 ?
(b) Consider the loop g : I → S 1 × S 1 given by
g(t) = (cos 4πt, sin 4πt) × (cos 2πt, sin 2πt).
Draw a lift starting at the origin of f ∗ g.
Problem 17. Prove that if X retracts onto a subset A ⊂ X then the fundamental group of A injects into that of X.
Problem 18. Prove that there is no retraction of B 2 onto S 1 .
Problem 19. Which of the capital letters A, C, D, E have nontrivial fundamental group? For the ones you think have nontrivial fundamental group, prove
it. (Hint: think about retractions.) Bonus: what is the fundamental group of
the capital letter B?
Problem 20.
(a) Suppose f : S 1 → S 1 is a continuous nulhomotopic map. Prove that f
has a fixed point. (Hint: f nulhomotopic means that it extends to a map
from B 2 .)
(b) Show that this f maps some point x to its antipode −x. (Hint: suppose
not and use this to define a homotopy F : S 1 × I → R2 r {0} between f
and the identity map—see the solutions to HW10.)
(c) Suppose g : S 1 → S 1 is the map which rotates the circle by an angle of π.
Prove that g does not extend to a continuous map from B 2 to S 1 .
Problem 21. Prove that there is no continuous map f : S 2 → S 1 sending
antipodes to antipodes; i.e., satisfying f (−x) = −f (x).