Download Problem 1: We denote the usual “Euclidean” metric on IRn by de : |x

Problem 1:
We denote the usual “Euclidean” metric on IRn by de :
q
|x| = x21 + · · · + x2n , de (x, y) = |x − y|.
(a) Show that each of the following two “d”’s are also metrics on IRn :
|x|1 = |x1 | + · · · + |xn |, d1 (x, y) = |x − y|1 .
|x|∞ = max {|xi | : i = 1, 2, . . . , n} , d∞ (x, y) = |x − y|∞
(b) Find the “best constants” for which the following inequalities hold for all x ∈ IRn :
m1 |x|1 ≤ |x| ≤ M1 |x|1 ,
(1)
m∞ |x|∞ ≤ |x| ≤ M∞ |x|∞ .
(c) Let n = 2 and draw sketches of B1 (0, 1) (the ball of radius 1 for d1 ) and B∞ (0, 1).
Draw another sketch which illustrates the inequalities of (b) when |x| = 1. Note
that, for example, x ∈ B1 (0, r) iff x/r ∈ B1 (0, 1). That is, for example, m|x|1 ≤ |x|,
amounts to Be (0, 1) ⊂ B1 (0, 1/m).
(d) Let U ⊂ IRn . Show that the following are equivalent: (i) U is open in (IRn , de ),
(ii) U is open in (IRn , d1 ), (iii) U is open in (IRn , d∞ ).
Problem 2:
Take S = [0, 1] and X = [0, 1] with the usual metric in Problem 6 of Section 1.2. Let
(F, ρ) be the complete metric space of that problem 6 and C be the subspace of F which
consists of continuous functions. Show that C is closed and hence complete. Construct a
sequence {fj }∞
j=1 in C such that ρ(fm , fn ) = 1 for all m, n ∈ {1, 2, . . .} and m 6= n. You
can do this with piecewise linear functions fj . Conclude that C is not totally bounded.
Problem 3:
Do Section 1.6, Problem 10, but write it up in a much more complete and pleasant to
read manner than the text does.
Problem 4: Let (X, d), (Y, ρ) be metric spaces.
(a) Let Y be complete. Let U be a subspace of X and f : U → Y be uniformly
continuous. Show that there is exactly one extension g of f to U which is continuous. That is, g : U → Y, g(x) = f (x) for x ∈ U, and g is continuous. Show
that g is uniformly continuous. (Hints: if xk ∈ U, xk → x ∈ U , show that {f (xk )}
is Cauchy in Y and define g(x) = limk→∞ f (xk ). Show this g has all the desired
properties.)
(b) Let Y be complete. Let {fj }∞
j=1 be an equi-uniformly continuous sequence of
functions, fj : X → Y. That is, for every > 0, there exists δ > 0 such that if
1
2
x, z ∈ X and d(x, z) < δ, then ρ(fj (x), fj (z)) < for all j = 1, 2, . . . . Show that
then
C = x ∈ X : lim fj (x) exists
j→∞
is closed in X and if f : C → Y is defined by f (x) = limj→∞ fj (x) for x ∈ C, then
f is uniformly continuous.
(c) Suppose that X is compact and Y is complete. In the notation of (b), show that
fj → f uniformly (i.e., in the metric of uniform convergence on X).
(d) Let Y be compact, X be separable and {fj }∞
j=1 be equi-uniformly continuous
as in (b). Using the compactness of Y, if {xk }∞
k=1 is a dense sequence in X,
construct a subsequence of {fj } whose values at x1 converge, then a subsequence
of that sequence whose values at x2 converge, etc, and by a diagonal argument,
a subsequence of the {fj } whose values at each xk converge. Conclude that {fj }
has a subsequence which converges at each point of X; moreover, if X is compact,
the convergence is in the metric of uniform convergence on X.
Problem 5: These are not at all hard (in contrast to Problem 4), and are mostly done in
the main text or answers of the book. But you’ll learn more if you work them out and
write them up.
T
(a) Let X be a set and Tα , α ∈ A, be a family of topologies on X. Show that α∈A Tα
is a topology on X.
(b) Let S be a family of subsets of X. Show that there is a topology T on X such
that S ⊂ T and if T̂ is another topology on X with the the property S ⊂ T̂ , then
T ⊂ T̂ . That is, T is the smallest topology containing S. It is called the topology
generated by S. Remember that I use A ⊂ B to mean the same as A ⊆ B. Hint:
Once you find one topology containing S, then (a) provides the smallest one.
(c) Suppose that S in (b) has the property that if U, V ∈ S and x ∈ U ∩ V, then there
exists W ∈ S such that x ∈ W and W ⊂ U ∩ V. Show that the topology generated
by S is then the set of all unions of sets in S. In this event, S is called a base for
the topology it generates.
(d) If (X, TX ), (Y, TY ) are two topological spaces and B is a base for TY , then f : X →
Y is continuous iff f −1 (B) ∈ TX for each B ∈ B.
(e) Show that if (Y, T ) is a topological space, X is a set, and fα : X → Y for α ∈ A,
then there is a smallest topology on X such that fα is continuous for all α ∈ A.
We will say that this topology on X is induced by the family fα .
(f) Let (X1 , T1 ), (X2 , T2 ) be topological spaces. Show that B = {U1 × U2 : Uj ∈ Tj } is
a base for the topology on X1 × X2 induced by the projections of X1 × X2 on its
factors, as given by P1 (x1 , x2 ) = x1 and P2 (x1 , x2 ) = x2 .
(g) Suppose now that (X, T ) is a topological space and Y is a set and f : X → Y.
Give an explicit description of the largest topology on Y for which f is continuous,
thus showing there is one.
3
Problems 6-9 are not hard and not long (I resisted saying “are short and easy” :)); we’re
just trying to give some exercises not entirely in the book. You are also learning a tad
more about the product topology and such, which we won’t do formally in lecture.
Problem 6: Let X, Y be topological spaces and f : X → Y be continuous. If Y is
Hausdorff, show that the graph G(f ) of f as given by
G(f ) = {(x, f (x)) : x ∈ X}
is a closed subspace of X × Y. Is the result true in general, without assuming that Y is
Hausdorff? (The topology on X × Y is given in Problem 5 (f).)
Remark: Of course, a function “is” its graph, but folks seldom really think that way.
Analysts tend to think that f does something to x which results in f (x). The function is
the “something” it does.
Problem 8: Let X, Y be topological spaces. Is the map
F :X ×Y →Y ×X
given by F (x, y) = (y, x) always a homeomorphism?
Problem 7: Let X be a topological space and E be a connected subspace of X. If A ⊂ X
satisfies E ⊂ A ⊂ Ē, show that A is connected. Conclude that closures of connected
sets are connected and connected components are connected. Show by example, that, in
contrast, components are not always open (we gave an example in class - just state it
cleanly). However, if X has only a finite number of distinct components, show that the
components of X are open.
Problem 8: Let X be a topological space. Show that X is connected exactly when there
are no continuous functions f : X → IR with the property f (IR) = {0, 1} .
Problem 9: Show that if f (x) = x17 + 8x6 + x − π, then 0 ∈ f (IR) (more generally, real
polynomials of odd degree have at least on real root). Show that the polynomial given
has a root in the interval [0,4].