Math 415 Review problems for Midterm
1. True or False: justify.
a. A ∪ B = A ∪ B
b. (A ∪ B)◦ = A◦ ∪ B ◦
c. The sequence {an } converges if and only if the sequence {a2n } converges.
d. Numbers of the form 3kn , k, n ∈ N are dense in the positive real numbers.
e. Let F be continuous on R. Then f −1 ((a, b)) is an open interval.
f. Suppose E is a finite dense subset of a metric space M. Then M is finite.
2. Show that X = (0, 1) and Y = (0, ∞) with the standard metric are homeomorphic
but not isometric.
3. Let A = {x ∈ (0, 1) : the base 5 expansion of x has only 2’s and 4’s. Show that
A is uncountable.
4. Let f : (X, dX ) :→ (Y, dY ) be a homeomorphism. Define d : M × M → R by
d(x1 , x2 ) = dY (f (x1 ), f (x2 )). Show that d is a metric on M which is equivalent to dX .
5. Show that f : (X, dX ) → (Y, dY ) is continuous if and only if f (A) ⊂ f (A), ∀A ⊂ X.
6. Show that any subset of Q with more that one point is disconnected.
7. Show there is no continuous bijection f : [0, 1) → R.
8. Show that the set S of all finite subsets of N is countable.
9. Let T : (M, d) → (M, d) be a strict contraction, that is there exists θ ∈ (0, 1) such
that
d(T x, T y) ≤ θd(x, y) .
Show that T has a unique fixed point if M is complete, that is every Cauchy sequence
converges. (Hint: Show the sequence defined by xn+1 = T xn , is Cauchy.)
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10. Let f ≥ 0 be continuous on [0, 1].
i. Show sup[0,1] f (x) = M < ∞.
ii. Show for any ε > 0 there exists a nonempty open interval Iε ⊂ (0, 1) such that
f > M − ε.
11. Construct a sequence {xn } ⊂ R such that the set of all subsequential limits is
exactly [0,1].
12. Let (M, d) be a connected metric space. Let ε > 0. By a ε-chain in the metric space (M, d) we mean a finite sequence of points x0 , x1 , .., xn of points of M s.t.
d(xi−1 , xi ) < ε for i = 1, 2, .., n. An ε-chain joining two points p and q of M is an
ε-chain x0 , x1 , .., xn s.t. x0 = p and xn = q. Show that any two points p,q in M can
be joined by an ε chain.
13. Let A and B be disjoint closed sets in a metric space (M, d). Show that there are
disjoint open sets U and V such that A ⊂ U and B ⊂ V .
Hint: Consider U = {x : d(x, A) < d(x, B)} and V = {x : d(x, B) < d(x, A)}.