Math 410 Final Exam
Instructions: Due Monday May 9. You may collaborate and use
outside resources. Make sure to cite all sources as outlined in the course
syllabus.
(1) Let X and Y be metric spaces. Assume that X is compact. Let
f : X → Y be a continuous bijection. Prove that f −1 : Y → X
is continuous.
(2) Prove that there is no continuous bijective map f : [0, 1] →
[0, 1] × [0, 1]. (Hint: use (1) and a connectivity argument.)
(3) Recall that if A and B are subsets of a metric space X, then
we define d(A, B) = inf{d(x, y) : x ∈ A, y ∈ B}.
(a) Exhibit two closed, connected subsets A, B ⊆ R2 which are
disjoint, but with d2 (A, B) = 0.
(b) Exhibit two closed subsets A, B ⊆ R which are disjoint, but
with d2 (A, B) = 0.
(4) Let X be a metric space. Let A be a countably infinite subset
of X. Prove that A is not connected.
(5) (a) Let f : R2 → R be a C 1 function. Prove that f is not
injective.
(b) Let f : R → R2 be a C 1 function. Prove that f is not
surjective.
(6) Let Q ⊆ Rn be a rectangle. Let (fn )n∈N be a sequence of
bounded real-valued functions on Q. Suppose that the sequence
(fn )n∈N converges uniformly to a function f : Q → R, i.e., suppose that limn→∞ supx∈Q |fn (x) − f (x)| = 0.
(a) Let x0 ∈ Q and assume that each of the functions fn is
continuous at x0 . Prove that f is continuous at x0 .
(b) Assume that each of the functions fn is integrable over
Q. Prove that f is integrable over Q.
(7) Let (X, d) be a compact metric space. Let h : X → X be an
isometry, i.e., d(h(x), h(y)) = d(x, y) for all x, y ∈ X. Prove
that h is bijective.
(8) Find a diffeomorphism g : R2 → R2 which preserves volume,
but which is not an isometry.
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