Download M2PM5 METRIC SPACES AND TOPOLOGY SPRING 2016 Exercise

M2PM5 METRIC SPACES AND TOPOLOGY
SPRING 2016
EXTRA PROBLEM SHEET
Exercise 1. Let (X, dX ) and (Y, dY ) be metric spaces. Define
dp : (X ⇥ Y ) ⇥ (X ⇥ Y ) ! R
p
(x1 , y1 ), (x2 , y2 ) 7 ! p dX (x1 , x2 )p + dY (y1 , y2 )p
A generalization of the Cauchy–Schwartz inequality, which you may assume without proof, is
Hölder’s inequality. This states that, if p 2 [1, 1) and q 2 [1, 1) are such that 1/p + 1/q = 1,
then:
n
n
n
⇣X
⌘1/p ⇣ X
⌘1/q
X
p
ai bi 
|ai |
|bi |q
i=1
i=1
i=1
for all a, b 2 R . Use this to show that dp is a metric on X ⇥ Y .
n
Exercise 2. Let X denote the set of all continuous functions from the interval [ 1, 1] to R.
Define:
d1 : X ⇥ X ! R
Z 1
(f, g) 7 !
|(f (t)
d1 : X ⇥ X ! R
(f, g) 7 ! sup |f (t)
g(t)| dt
g(t)|
t2[ 1,1]
1
Both (X, d1 ) and (X, d1 ) are metric spaces. (You do not need to prove this.) Consider the
function
ev0 : X ! R
f 7 ! f (0)
Is ev0 a continuous function on (X, d1 )? Is ev0 a continuous function on (X, d1 )?
1
M2PM5 METRIC SPACES AND TOPOLOGY
SPRING 2016
PROBLEM SHEET 1
Exercise 1. Let p be a prime number. Define a function d : Z ⇥ Z ! R by
(
0 if m = n
d(m, n) = 1
if m 6= n, where m n = pr 1 q with q 2 Z not divisible by p.
r
Show that d is a metric on Z.
Exercise 2. Let C([a, b]) denote the set of continuous functions from [a, b] to R, and let
C 1 ([a, b]) denote the set of di↵erentiable functions f : [a, b] ! R such that f 0 is continuous.
Let:
d1 (f, g) = sup |f (x) g(x)|
x2[a,b]
1
This defines a metric on both C([a, b]) and C ([a, b]).
(1) Consider the map:
Int : C([a, b]), d1 ! C 1 ([a, b]), d1
Z x
f 7!
f (t) dt
a
Is Int continuous?
(2) Consider the map:
Diff : C 1 ([a, b]), d1 ! C([a, b]), d1
f 7! f 0
Is Diff continuous?
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PROBLEM SHEET 1
Exercise 3. Let (X, d) be a metric space and A be a subset of X. Show that x 2 @A if and
only if, for all ✏ > 0, we have that B✏ (x) \ A and B✏ (x) \ (X \ A) are both non-empty.
Exercise 4. Let (X, d) be a metric space and A be a subset of X. For x 2 X, define
d(x, A) = inf{d(x, a) : a 2 A}
Show that:
(1) d(x, A) = 0 if and only if x 2 A.
(2) for all y 2 X, d(x, A)  d(x, y) + d(y, A).
(3) the map x 7! d(x, A) defines a continuous function from X to R.
M2PM5 METRIC SPACES AND TOPOLOGY
SPRING 2016
PROBLEM SHEET 2
Exercise 1
Let (X, d) be a metric space.
(1) Show that, given any two distinct points x, y 2 X there exist open sets U , V in X with
x 2 U , y 2 V and U \ V = ;.
(2) Suppose that B is a bounded subset of X and that C ⇢ B. Show that C is bounded,
and that diam C  diam B.
Exercise 2
Let f : R ⇥ R ! R be defined by:
f (x, y) =
(
xy
x2 +y 2
0
(x, y) 6= (0, 0)
(x, y) = (0, 0
Show that the restriction of f to the subset R ⇥ R \ {(0, 0)} is continuous. Is f continuous?
Exercise 3
(1) Let (X, d) be a metric space and let A1 , . . . , Am be subsets of X. Show that:
✓ i=m
[ ◆ i=m
[
Ai =
Ai
i=1
i=1
(2) Let (X, d) be a metric space and let Ai , i 2 I, be subsets of X. Show that:
✓ i=m
\ ◆ i=m
\
Ai ✓
Ai
i=1
i=1
(3) Give an example of a metric space (X, d) and subsets A, B of X such that:
A \ B 6= A \ B
Exercise 4
Compute the closure of each of the following sets in R:
(1) [1, 1)
(2) R \ Q
n
(3) { n+1
: n 2 N}
1
(4) { n : n 2 N, n 2} [ {0, 1, 2}
1
2
PROBLEM SHEET 2
Exercise 5
The Cantor set C is defined as follows. Let C0 = [0, 1] and, for n 0, let Cn+1 be the set
obtained from Cn by taking each maximal closed interval I contained in Cn and removing from
I the open interval that forms the middle third of I. Set:
\
C=
Cn
n 0
n
(1) Show that Cn is a disjoint union of 2 closed intervals;
(2) Show that C is closed.
(3) Show that C is non-empty.
(4) Optional: show that C is uncountable.
(5) Optional: show that every point in C is an accumulation point.
(6) Optional: show that C has empty interior.
The Cantor set is an example of an uncountable set with Lebesgue measure zero. Items (2)
and (5) here show that C is a so-called perfect set. Item (6) shows that C is nowhere dense in
R.
M2PM5 METRIC SPACES AND TOPOLOGY
SPRING 2016
PROBLEM SHEET 3
Exercise 1. Suppose that X is an infinite set equipped with the cofinite topology.
(1) Let A ⇢ X be a finite set. Compute A.
(2) Let A ⇢ X be an infinite set. Compute A.
Exercise 2. Let X be a non-empty set and let T1 and T2 be topologies on X. Must T1 \ T2 be
a topology on X? Must T1 [ T2 be a topology on X?
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2
PROBLEM SHEET 3
Exercise 3. Let A be a subset of the topological space X. Show that @A = A \ X \ A.
Exercise 4. Let X and Y be topological spaces. Let:
B = {U ⇥ V : U is open in X and V is open in Y }
The product topology on X ⇥ Y is
T = {Z ⇢ X ⇥ Y : Z is a union of elements of B}
This is a topology on X ⇥ Y . (You do not need to show this.)
(1) Show that the projection maps:
p1 : X ⇥ Y ! X
(x, y) 7! x
p2 : X ⇥ Y ! Y
(x, y) 7! y
are continuous, where X ⇥ Y is given the product topology.
(2) Let Z be a topological space. Show that a map f : Z ! X ⇥ Y is continuous if and
only if p1 f and p2 f are continuous.
M2PM5 METRIC SPACES AND TOPOLOGY
SPRING 2016
PROBLEM SHEET 4
Exercise 1.
(1) Let X be a non-empty set equipped with the discrete topology. Show that X is compact
if and only if X is finite.
(2) Let X be a topological space, and let Y and Z be compact subsets of X. Show that
Y [ Z is compact.
Exercise 2. Let X be a compact topological space and let f : X ! Y be a continuous map
of topological spaces. Show that f (X) is compact.
(This would make an excellent exam question. If you can do it, then you are probably getting
the hang of compactness.)
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PROBLEM SHEET 4
Exercise 3. Let X be a compact topological space. Suppose that, for each n 2 N, Vn is a
closed non-empty subset of X and that:
Show that:
V0 ◆ V1 ◆ V2 ◆ · · ·
\
n 0
Vn 6= ?
Is this statement true without the compactness hypothesis?
Exercise 4. Suppose that X is a topological space and that A, B are connected subsets of X
such that A \ B 6= ?. Show that A [ B is connected.
M2PM5 METRIC SPACES AND TOPOLOGY
SPRING 2016
PROBLEM SHEET 5
Exercise 1. Let (X, T ) be a topological space. Let 1 be an object not in X, and set:
X̌ = X [ {1}
Ť = T [ V [ {1} : V ✓ X such that X \ V is compact and closed
(1) Show that Ť is a topology on X̌.
(2) Show that the topological space (X̌, Ť ) is compact.
(3) Show that the topological space (X̌, Ť ) contains (X, T ) as a subspace.
(4) Suppose that X = R with the usual topology. What is (X̌, Ť )?
(5) Suppose that X = R2 with the usual topology. What is (X̌, Ť )?
The space (X̌, Ť ) is called the one-point compactification of X.
Exercise 2.
(1) Show that the annulus {(x, y) 2 R2 : 1 < x2 + y 2 < 2} is path connected.
(2) Show that the region {(x, y) 2 R2 : x < 0 or x > 1} is not path-connected.
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PROBLEM SHEET 5
Exercise 3. Consider the topologist’s sine curve T , which is the subspace of R2 defined by:
T = {(0, 0)} [ (x, sin x 1 ) : x 2 (0, 1)
(1) Draw a picture of T .
(2) Show that T is connected.
(3) Show that T is not path-connected.
M2PM5 METRIC SPACES AND TOPOLOGY
SPRING 2016
PROBLEM SHEET 6
Exercise 1. Consider the following functions fn : [0, 1] ! R. In which case does the sequence
(fn )n 1 converge uniformly on [0, 1]?
x
(1) fn (x) = 1+nx
xn
(2) fn (x) = 1+x
n
2
(3) fn (x) = nx(1 x2 )n
Exercise 2. Construct functions fn : R ! R such that none of the fn is continuous at 0 2 R
but that (fn )n 1 converges uniformly on R to a continuous function.
1
2
PROBLEM SHEET 6
Exercise 3. Let X be the metric space [1, 1), considered as a subspace of R. Let f : X ! X
be the map x 7! x + x 1 . Show that:
(1) X is complete;
(2) |f (x) f (y)| < |x y| for all x, y 2 X;
(3) f has no fixed point.
Exercise 4. We say that a metric space X is totally bounded if and only if for each ✏ > 0 there
exist finitely many points x1 , . . . , xN 2 X such that:
B✏ (x1 ), . . . , B✏ (xN )
is a cover of X. Show that a metric space X is compact if and only if it is complete and totally
bounded.
(Hint: for one direction, show that complete and totally bounded implies sequentially compact.)
M2PM5 METRIC SPACES AND TOPOLOGY
SPRING 2016
PROBLEM SHEET 7
Exercise 1. Give an explicit homotopy between the following paths in R2 :
f : [0, 1] ! R2
g : [0, 1] ! R2
t 7 ! cos ⇡t, sin ⇡t
t 7 ! cos ⇡t,
sin ⇡t
Exercise 2. We say that a subspace D of Rn is star-shaped if and only if there exists a point
x0 2 D such that, for all x 2 D, the straight line segment from x0 to x lies entirely within D.
Let D be a star-shaped subspace of Rn .
(1) Show that D is path-connected.
(2) Show that ⇡1 (D, x0 ) = {1}.
When combined with the next exercise, this shows that D is simply-connected.
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PROBLEM SHEET 7
Exercise 3. This exercise shows that, up to isomorphism, the fundamental group ⇡1 (X, x0 )
depends only on the path-component of X that contains x0 .
Let X be a topological space, let x0 and x1 be points in X, and let : [0, 1] ! X be a path
1
from x0 to x1 . Let 1 : [0, 1] ! X be the reverse path:
(t) := (1 t). Define a map
M : ⇡1 (X, x0 ) ! ⇡1 (X, x1 )
(1)
(2)
(3)
(4)
(5)
Show that M
Show that M
Show that M
Show that if
Suppose that
M = M 0?
[f ] 7 ! [ ⇤ f ⇤
1
]
is well-defined. That is, show that if f ⇠ g then ⇤ f ⇤ 1 ⇠ ⇤ g ⇤ 1 .
is a homomorphism of groups.
is an isomorphism. (Hint: what is (M ) 1 ?)
⇠ 0 then M = M 0 .
and 0 are both paths in X from x0 to x1 . Is it always the case that
M2PM5 METRIC SPACES AND TOPOLOGY
SPRING 2016
PROBLEM SHEET 8
Exercise 1. Consider the map p : R ! S 1 given by t 7! cos 2⇡t, sin 2⇡t . Show that this is
a covering map. That is, exhibit an open cover {Ui : i 2 I} of S 1 such that for each i 2 I,
p 1 (Ui ) is a disjoint union:
a
p 1 (Ui ) =
V↵
↵
where p|V↵ gives a homeomorphism from V↵ to Ui .
Exercise 2. Path lifting. Let : I ! S1 be a path in S 1 from x0 to x1 and let x̃0 2 R be such
that p(x̃0 ) = x0 . We will show that there is a unique path ˜ : I ! R such that ˜ begins at x̃0
and that p ˜ = . To do this:
(1) Show that there exist 0 = t0 < t1 < · · · < tN = 1 such that ([ti 1 , ti ]) lies entirely
within one of the open sets Uj that you constructed in Exercise 1.
(Hint: Lebesgue number!)
(2) Suppose that we have constructed the lifted path ˜ on the subinterval [0, ti 1 ]. Explain
how to extend this to get the lifted path ˜ on the subinterval [0, ti ].
(Hint: use the fact that p is a covering map.)
(3) Show that the lifted path ˜ : I ! R thus constructed is the unique path ˜ : I ! R such
that ˜ begins at x̃0 and that p ˜ = .
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PROBLEM SHEET 8
Exercise 3. Homotopy lifting. Suppose that Y is a topological space, that F : Y ⇥ I ! S 1 is
a continuous map, and that f˜: Y ! R is such that p f˜ = F |Y ⇥{0} . We will show that there
is a unique continuous map F̃ : Y ⇥ I ! R such that F̃ |Y ⇥{0} = f˜ and that p F̃ = F .
(1) Show that, for each y0 2 Y , we can find a finite open cover V1 , . . . , VM of {y0 }⇥I ⇢ Y ⇥I
such that each Vj is connected and F (Vj ) lies inside one of the sets Ui constructed in
Exercise 1.
(2) Write Vy0 = V1 [ · · · [ VM . Arguing as in Exercise 2, show that there is a unique map
F̃y0 : Vy0 ! R such that p F̃y0 = F |Vy0 and that F̃ (y0 , 0) = f (y0 ).
(3) Show that Vy0 contains N ⇥ I for some open neighbourhood N of y0 in Y . Deduce that
the lifts F̃y0 agree for varying y0 2 Y .
This shows that defining F̃ : Y ⇥ I ! R by F̃ (y, t) = F̃y (y, t) gives the unique continuous map
F̃ : Y ⇥ I ! R such that F̃ |Y ⇥{0} = f˜ and that p F̃ = F .
M2PM5 METRIC SPACES AND TOPOLOGY
SPRING 2016
MOCK FINAL EXAM QUESTIONS
Abstract. There are too many questions here – the final itself will have four questions. Also
question 6 is probably too long to be a good exam question. But nonetheless these should give
you some useful practice ahead of the final exam. I will post solutions on the class website after
the Easter holiday.
(1) Give examples of the following, or prove that no such examples exist.
(a) A metric space X and open subsets Ui , i 2 I, of X such that \i2I Ui is not open.
(b) An unbounded metric space X and a sequence (xn )n 1 of points of X such that
(xn ) has no convergent subsequence.
(c) A compact metric space X such that X is not complete.
(d) A topological space X, a point x0 in X and a loop : [0, 1] ! X based at x0 such
that is homotopic to , where is the reversed loop: (t) = (1 t).
(e) Topological spaces X and Y and a continuous map f : X ! Y such that X is
compact, Y is Hausdor↵, and f (X) is not closed.
(f) Topological spaces X and Y , an open set U ⇢ X, and a continuous map f : X ! Y
such that f (U ) is not open.
(2) (a) Suppose that X is a topological space and that A, B are connected subsets of X
such that A \ B 6= ?. Show that A [ B is connected.
(b) Consider the subspace T of R2 defined by:
T = {(0, 0)} [ (x, sin x 1 ) : x 2 (0, 1)
Is T connected? Is T path-connected?
(3) Suppose that X is a set and that fn : X ! R, n 2 {1, 2, 3, . . .}, and f : X ! R are
maps.
(a) Define what it means for the sequence (fn )n 1 to converge to f pointwise. Define
what it means for the sequence (fn )n 1 to converge to f uniformly.
(b) Prove or disprove: if (fn )n 1 converges to f uniformly then (fn )n 1 converges to f
pointwise.
(c) Prove or disprove: if (fn )n 1 converges to f pointwise then (fn )n 1 converges to f
uniformly.
Suppose now that X is a compact topological space, that each map fn : X ! R is
continuous, and that fn+1 (x)  fn (x) for all x 2 X and all n. Suppose further that
(fn )n 1 converges to f pointwise and that f is continuous. Show that (fn )n 1 converges
to f uniformly.
(4) (a) Let X be a metric space. Define what it means for a map f : X ! X to be a
contraction.
(b) Let X be a complete metric space. Let k be a positive integer and let f : X ! X
be a map such that the k-fold composition”
f (k) = f
|
f
··· f
{z
}
k times
1
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MOCK FINAL EXAM QUESTIONS
is a contraction. Show that f has a unique fixed point in X. Prove or disprove: or
any x 2 X, the sequence f (n) (x) n 1 converges to the fixed point.
(5) Let:
B = f : [0, 1] ! R : f is bounded
C = f : [0, 1] ! R : f is continuous
C 1 = f : [0, 1] ! R : f is di↵erentiable and f 0 2 C .
(a) Show that
d(f, g) = sup |f (t)
g(t)|
t2[0,1]
defines a metric on B.
(b) Prove or disprove: C is closed in B.
(c) Consider the maps:
D : C1 ! C
f 7 ! f0
Prove or disprove:
(i) D is continuous;
(ii) I is continuous.
I: C ! C
Z x
f 7!
f (t) dt
0
(6) (a) Let f : A ! B be a continuous map between topological spaces, let a0 2 A, and
let b0 = f (a).
(i) Define the fundamental group ⇡1 (A, a0 ).
(ii) Define the map f? : ⇡1 (A, a0 ) ! ⇡1 (B, b0 ) and prove that it is well-defined.
(b) Let X and Y be topological spaces, and let pX : X ⇥ Y ! X and pY : X ⇥ Y ! Y
be the projection maps. We consider X ⇥Y as a topological space with the product
topology.
(i) What does it mean for a subset U ✓ X ⇥ Y to be open?
(ii) Show that pX and pY are continuous.
(iii) Show that, for a topological space Z, F : Z ! X ⇥ Y is continuous if and
only if pX F and pY F are continuous.
(iv) Hence, or otherwise, show that
⇡1 X ⇥ Y, (x0 , y0 ) ⇠
= ⇡1 (X, x0 ) ⇥ ⇡1 (Y, y0 )
as groups.
(v) Show that (S 1 )n is homeomorphic to (S 1 )m if and only if n = m.