Download X → Y must be constant. .... Let T

If X is a connected space and Y a totally disconnected space that every continuous
map f : X → Y must be constant.
....
Let T be a set and B a collection of subsets of T . Show that if B is closed under
finite intersections, then the collection of all unions of sets in B forms a topology
on T .
...
from Viro: 4.9 Find a metric space and two balls in it such that the ball with the
smaller radius contains the ball with the bigger one and does not coincide with it.
4.17. Prove that every ball in a normed space is a convex set symmetric with respect
to the center of the ball.
4.32. ... are min{d_1, d_2} , d_1 d_2, d_1/d_2 metrics 4.33. if f is increasing,
f(0)=0, and f is subadditive then f(d(.,.)) is a metric
1
2
Exercises on Topology and Metric Spaces.
Exercise 1. (continuity)
[A] A function between metric spaces is continuous iff it commutes with limits. [In
a metric space the topology is described by limits! ]
[B] [Continuity is local! ] A map on the union of two closed sets is continuous iff the
two restrictions are continuous. A map on the union of two open sets is continuous
iff the two restrictions are continuous.
[C] A map between metric spaces is continuous iff the restriction to every compact
subset is continuous.
[D] If f is a continuous function then for any subset S in the domain we have
f (clsS) ⊂ clsf (S). How about the interior? Can one characterize continuity in
terms of the closure or interior operators?
Exercise 2. (compactness)
(1)
(2)
(3)
(4)
(5)
(6)
A compact metric space is complete.
A bounded closed set (interval) in the rationals is not compact.
A bounded closed set (closed ball) in l2 is not compact.
The intersection of a family of compact spaces is compact.
A metric space is compact iff every continuous real function on it is bounded.
The derived set (i.e., the set of accumulation points) of a compact metric
space is compact.
(7) A metric space is compact iff every decreasing sequence of closed non-empty
sets has a non-empty intersection.
Exercise 3. (connectedness)
A space is connected iff all the continuous maps to the discrete space with two
points are constant.
Exercise 4. [base, sub-base, and local base of a topology; to generate a topology; first and second countable; finer and coarser topologies] In a metric space
the topology, is defined via the balls. Generalizing this property of the collection
of open balls to any topological space one introduces the notion of a base: In a
topological space (X, T ) a base of the topology T is a collection B of open sets
such that for every point x in X and every open neighbourhood U 3 x there is a
B ∈ B such that x ∈ B ⊂ U . Thus the topology induced by a metric has as base
the collection of open balls. In a similar way extracting a property of the collection
of balls that contain a fixed point we arive at the notion of a local base: A local
base for the open neighbourhoods of a point x, or a local base of x, is a collection
Bx of open neighourhoods of x such that for for any U , x ∈ U ∈ T , there exist
B ∈ B, such that B ⊂ U .
In a topological space (X, T ) a subcollection B ⊂ T is a base for the topology T
iff every open set is a union of sets from the base. (One can say that the topology
is the closure of the base under arbitrary unions, or that the topology is generated
from the base via arbitrary unions. Often this is taken as the definition of base.)
3
B is a base for the topology T iff for every x ∈ X the collection Bx = {B ∈ B : x ∈
B} is a local base at x.
In a set X a collection of subsets B is a base for a topology on X iff B is a cover of
X and if for every two sets B 0 , B 00 ∈ B and every point x ∈ B 0 ∩ B 00 there is B ⊂ B
such that x ∈ B ⊂ B 0 ∩ B 00 .
(Note that the last statement shows that not every collection of subsets is a base.
We would like to start with an arbitrary collection of subsets of X and produce
a base of a topology on X. One checks the following:) Take any collection S of
subsets of X that covers X. The collection of all finite intersections of elements of
S is a base for a topology T on X. This topology is the smallest one containing S
and we say that S is a sub-base for T or that T is generated by S. We could
write this as T = T (S).
To check continuity it is enough to check it on bases. Let f : X1 → X2 be a function
between topological spaces. Let B1 and B2 be bases of the topologies on the source
and the target. The f is continuous iff for every for every B2 ∈ B2 , there is a
B1 ∈ B1 , such that f (B1 ) ⊂ B2 , or equivalently, such that B1 ⊂ f −1 (B2 ). (In a
similar way one can define continuity at a point x by comparing how f transforms
the local bases at x and f (x).)
Lets have two topologies T1 and T2 on X. Let Bi be a base of Ti . We can compare
the two topologies via their bases, i.e., T1 ⊂ T2 iff for every point x and every B1 ,
x ∈ B1 ∈ B1 , there is B2 , x ∈ B2 ∈ B2 , such that B2 ⊂ B1 . (The bigger or stronger
topology T2 is the one with the finer collection of open sets, hence is also called the
finer topology, and the smaller or weaker topology T1 is the one with the coarser
collection of open sets, hence also called coarser.)
Consider the possible topologies of a finite set. Take a set with one, two, or three
elements. Can you go further with an explicit description?
Lets have a function between two sets f : X → Y . If we have a topology TY on the
target we can consider the topology T (f −1 (TY )) generated by the preimages of open
sets. This is the weakest, or coarsest, topology on X making f continuous. If we
have a topology TX on the source we can consider the topology T (f (TX )) generated
by the images of open sets. This is the strongest, or finest, topology on Y making
f continuous. This can be generalized to families of maps. http://en.wikipedia.
org/wiki/Initial_topology http://en.wikipedia.org/wiki/Final_topology
The induced topology on a subset X ⊂ Y of a topological space (Y, T ) is the
weakest topology on X making the inclusion continuous. The other characterization
of the induced topology is T |X = {U ∩ X : U ∈ T }.
Consider now a product X1 × X2 . It is equipped with the cannonical projections
pi : ×j=1,2 Xj → Xi that “forgets all but the i-th coordinate”. If we have topological
spaces (Xi , Ti ) the product topology is the weakest topology on the product
making the projections continuous. The equivalent characterization is that it is the
topology on the product that is generated by the cylindrical sets p−1
1 (U1 ) = U1 ×X2 ,
p−1
(U
)
=
X
×U
,
where
U
∈
T
.
(In
the
plane
what
are
the
cylinders
on intervals?
2
1
2
i
i
2
Draw examples of cylindrical sets. Intersect two cylinders to obtain a rectangle in
the plane. In particular the squares in the plane are balls of which metric on the
plane?) The product topology is defined in the same way for arbitrary products.
4
Let n = {1, . . . , n} and view the Cartesian product Rn as the space of maps Rn =
{x : n → R : i 7→ x(i)}. A sequence of points N 3 j 7→ xj = (xj (1), . . . , xj (n)) converges to x = (x(1), . . . , x(n)) ∈ Rn coordinatewise, or pointwise, if limj→∞ xj (i) =
x(i) for every i ∈ n. This defines the same topology as the product topology on Rn .
More generally, given a metric space Y and an index set X the space of functions
Y X = {X → Y } when equipped with the topology of pointwise convergence is the
same as the product space Y X = ×x∈X Y equipped with the product topology.
If X is a totaly ordered set, i.e., any two points are comparable, we can define the
order topology as the topology generated by the open intervals or the open rays.
The order topology on the reals coincides with the metric topology coming from
the absolute value.
The axioms of compactness characterize a topological space in terms of the cardinality of a “small” base. A space is first-countable if every point has a countable
base. A space X is second-countable if there is a countable base for the topology
of X. Every metric space is first countable. The real line is second countable and
also every finite product of the reals, i.e., every finite dimensional real vector space,
is second countable.
A first-countable space is quite close to a metric space and the topology is described
in terms of sequences and limits. A point is in the closure of a set A iff there is
a sequence in A converging to the point. A function with souce a first-countable
space is continuous iff it commutes with limits. (But there are sequentially compact
first-countable spaces which are not compact.) http://en.wikipedia.org/wiki/
First-countable_space
What is a compactly generated space? http://en.wikipedia.org/wiki/Compactly_
generated_space
Exercise 5. (Cantor set)
The Cantor set is a not empty closed subset of the reals.
The Cantor set has cardinality equal to the continium, i.e., the cardinality of the
reals. [Hint: what is the ternary expansions of the points in the Cantor set?]
The Cantor set is nowhere dense ???
The Cantor set has Lebesgue measure zero.
The Cantor staircase function is not Riemann integrable but its Lebesgue integral
exists.
[universality of the Cantor set] Every compact metric space is a continuous image
of the Cantor set.
Exercise 6. (two metrics on the same space)
If d1 and d2 are metrics on X then d1 + d2 is also a metric. If d is a metric and c, a
positive real number, then c d is a metric.
Let X be a set with two metric functions d1 and d2 . Let the corresponding topologies be T1 and T2 . If there is a constant c, 0 < c < ∞, such that d1 ≤ c d2 (of course
by this we mean pointwise, i.e., d1 (x, y) ≤ c d2 (x, y) for arbitrary points x, y ∈ X)
we will say that d1 is majorized by d2 (or that d2 is minorized by d1 ). Note that
one constant does the job for all pairs of points. If d1 is majorized by d2 what is
5
the relation between the two topologies (are they comparable, and if yes, which is
finer and which coarser)?
Any two metrics on the same space can be majorized by a third metric. But it
is not true that they can be minorized. [Hint: The sum majorizes. For a counter
example take the reals with d1 the usual metric, let f exchange 0 and 1 and fix
every other real number and take d2 (x, y) = d1 (f (x), f (y)).]
Two metrics on the same space are called (strongly) equivalent if they majorize
each other, i.e., if there are constants c0 , c00 , c0 , c00 > 0, such that c0 d1 ≤ d2 ≤ c00 d1 .
Two metrics on the same space are called (topologically) equivalent if they induce
the same topology. How are the strongly equivalent and topologically equivalent
metrics related?
If two metrics are strongly equivalent (i.e., if there are constants c0 , c00 , c0 , c00 > 0,
such that c0 d1 ≤ d2 ≤ c00 d1 ) then a set is bounded in one iff it is bounded in the
other.
Consider a metric space (X, d). Define new functions d0 = min(1, d) and d00 =
d/ (1 + d), i.e., d0 (x, y) = min(1, d(x, y)) and d00 (x, y) = d(x, y)/ (1 + d(x, y)) for
any pair of points x, y ∈ X. Both are metrics on X. Are they equivalent (strongly
or topologically)? Are they bounded metrics on X? Give examples showing that
the property of being bounded is not preserved when a metric is changed to a
topologically equivalent metric.
Exercise 7. (metric topology)
A metric space is Hausdorff. ...
A metric space is first-countable. ....
Exercise 8. (playing with metrics)
Given two points x1 6= x2 in a metric space, and two real numbers s1 6= s2 , define a
function f on the metric space such that f (x1 ) = s1 and f (x2 ) = s2 . [Hint: define
f in terms of the metric.]
No bounded metric space can be a contraction on itself.
There is a metric space consisting of four points that cannot be embedded in the
3-dimensional Euclidean space.
Define a distance in the plane so it becomes isometric to the line. [Hint: Use
Cantor’s proof of the bijection between the two.]
If f (x) = x(1 − |x|)−1 then d(x, y) = |f (x) − f (y)| is a distance on (−1, 1) making
it into a space isometric with the reals.
Exercise 9. (distance between sets) Let A and b be a subset and a point in a
metric space. Define d(A, b) = inf a∈A d(a, b).
The point b is in the closure of A iff d(A, b) = 0.
The distance from a fixed A to the other points is uniformly continuous. [Hint:
Show that for all points u and w we have the inequality d(A, v) − d(A, w) ≤ d(v, w),
or equivalently d(A, v) ≤ d(B, w) + d(v, w). First note that for all x in S, d(x, v) ≤
d(x, w) + d(v, w). Then take the g.l.b. for x ∈ S on the left, and follow that by
taking the g.l.b. of d(x, w) + d(v, w) for x ∈ S on the right.]
6
Let A and B be closed and disjoint. The the function x 7→ d(A, x)/ (d(A, x) + d(B, x))
is continuous with values between 0 and 1 and takes the value 0 on A and 1 on B.
Let A and B be subsets in a metric space. Define d(A, B) = inf a∈A, b∈B d(a, b). If
the sets are closed and one of them is compact then there are elements a0 ∈ A and
b0 ∈ B such that d(A, B) = d(a0 , b0 ).
Exercise 10. (the operations in a normed vector spaces are continuous) Let X be
a normed vector space. For s ∈ R, x ∈ X, andU ⊂ X denote sU = {su : u ∈ U }
and x + U = {x + u : u ∈ U }.
If U is open then x + U and sU are open.
Addition and scaling are continuous operations.
Exercise 11. (linear operators between normed vector spaces) A linear mapL :
E → F between normed vector spaces is called bounded if there is a number
C > 0 such that kL(x)k ≤ C kxk for all x ∈ E. (Note that a bounded operator is
not a bounded function on the whole space but is bounded on the unit ball.) The
smallest such C is called the operator norm is and is denoted ..........
A linear function between two normed vector spaces is continuous iff it is continuous
at zero.
Let E and F be normed vector spaces, and let L : E → F be a linear map. (a)
Assume that there is a number C > 0 such that kL(x)k ≤ C kxk for all x ∈ E.
Show that L is continuous. (b) Conversely, assume that L is continuous at 0. Show
that there exists such a number C.
A linear map from a finite dimensional space is continuous.
Let L : E → F be a continuous linear map. Show that the values of L on the closed
ball of radius 1 are bounded. If r is a number > 0, show that the values of L on any
closed ball of radius r are bounded. (The closed balls are centered at the origin.)
Show that the image under L of a bounded set is bounded.
kL(x)k
. Show that the
kxk
continuous linear maps of E into F form a vector space, and that the above is a
norm on this vector space.
Let L be a continuous linear map, and let kLk = supx
Exercise 12. (normed vector spaces)
If k.k1 and k.k2 are two norms show that k.k1 + k.k2 and maxi=1,2 (k.k1 , k.k2 ) are
norms.
If k.k is a seminorm then the set of points for which the seminorm vanishes is a
vector subspace.
What is the relation between the norms k.k∞ , k.k1 , and k.k2 on the space of continuous functions on [0,1]. Are they equivalent?
The Euclidean norm is not isometric to the L1 norm ....
Give an example of a vector space with two norms, and a subset of the vector space
such that it is bounded for one norm but not for the other.
7
Give an example of a sequence in C 0 ([0, 1]) which is L2 -Cauchy but not Cauchy in
the sup norm. Is this sequence L1 -Cauchy? If it is, can you construct a sequence
which is L2 -Cauchy but not L1 -Cauchy? Why?
The unit interval [0, 1] cannot be represented as a countable disjoint union of closed
sets.
The real line is not a continuous image of [0, 1).
The real line is homeomorphic to (0, 1).
Let fn (x) = xn , and view {fn } as a sequence in C 0 ([0, 1]). Show that fn approaches
0 in the L1 -norm and the L2 -norm, but not in the sup norm.
Show that the space C 0 ([0, 1]) is not complete for the L2 -norm.
Consider a metric space (X, d). Let B(X) be the space of bounded real valued
functions on X with the sup norm. Denote fx (y) = d(x, y). Fix a point a ∈ X and
let gx = fx − fa . Then x 7→ gx is an isometric embedding of X into B(X). Note
that d(x, y) = kfx − fy k. (... all metric spaces are subsets of normed spaces....)
........................
On a finite dimensional vector space, two sup norms with respect to two different
bases are equivalent.
Give an example of a vector space with two norms, and a subset of the vector space
such that it is bounded for one norm but not for the other.
Let A be a dense supspace of a metric space X. If every Cauchy sequence in A has
a limit in X show that X is complete.
Let E be a complete normed vector space and let F be a subspace. Show that the
closure of F in E is a subspace. Show that this closure is complete.
Show that the closure of a convex set is convex.
Let a < b be numbers, and let C 0 ([a, b]) be the space of continuous functions on
[a, b]. Let Iab : C 0 ([a, b]) → R be the integral. Is Iab continuous: (a) for the L1 -norm;
and (b) for the L2 -norm. Prove your assertion.
A continuous linear map is uniformly continuous.
Let E = Rk and let S be a closed subset of Rk . Let v ∈ Rk . Show that there exists
a point w ∈ S such that d(S, v) = |w − v|. [Hint: Let B be a closed ball of some
suitable radius, centered at v, and consider the function x → |x—v| for x ∈ B ∩ S.]
Any two norms on a finite vector space are topologivally equivalent.
Let K be a compact metric space. Let f : K → K be a continuous expanding map.
Then f is injective, surjective, and the inverse is continuous.
F The interval [0, 1) is not complete with respect to the standard metric d(x, y) =
|x − y|. Find a metric D on this interval that is topologically equivalent to d but
the interval is complete with respect to D. [Gelbaum; p17]
F If f : [0, 1] → A × B is a homeomorphism then one of A or B is a singleton.
[Gelbaum; p17]
F Is d(x, y) = | arctan x − arctan y| a metric on R? Is R complete with respect to
this metric? [Gelbaum; p18]
8
F Are R\Q and R\Q ∩ (0, 1) homeomorphic? [Gelbaum; p18]
F A nonempty, countable, and compact subset K of a complete metric space X
contains an isolated point. [Gelbaum; p18]
F If a sequence of closed subsets of Rn constitute a cover of Rn then the union of
their interiors is dense in Rn . [Gelbaum; p18]