Download TOPOLOGY PROBLEMS FEBRUARY 23—WEEK 1 1

TOPOLOGY PROBLEMS
FEBRUARY 23—WEEK 1
1. Topological spaces
1. Consider the set {1, 2, 3} with ⌧ = {;, {1}, {1, 2}, {3}, {1, 2, 3}}.
Can you add a single set to ⌧ to make it a topology?
2. Define a topology on R as follows:
⌧ = {A ✓ R | R \ A a finite set} [ {;}
(i) Show that (R, ⌧ ) is a topological space.
(ii) Show that (R, ⌧ ) is not metrizable.
3. Define a topology on R as follows:
⌧ = {(a, 1) | a 2 R [ {±1}}
where we take (1, 1) to be ;.
(i) Show that (R, ⌧ ) is a topological space.
(ii) What is the closure of the set {0}?
(iii) Does the space have a base B ( ⌧ ?
(iiv) Show that (R, ⌧ ) is not metrizable.
4. Consider the integers Z, and say that a set A ✓ Z is open if it is
empty or if for every a 2 A, there is some nonzero b 2 N so that
{a + bn | n 2 Z} ✓ A
That is, a nonempty set A is open if every element is in an arithmetic
progression contained in A. Show that (Z, ⌧ ) is a topological space.
[Note: While still a student at university, Furstenberg defined this
topology and used it to (re)prove there are infinitely many prime numbers.]
2. Continuous functions, subspaces
5. Suppose f : X ! Y is a continuous map. Show that the inverse
image of any closed set in Y is closed in X.
1
2
WEEK 1
6. Show that a map f : X ! Y between topological spaces X and Y
is continuous if and only if f [Ā] = f [A].
7. Show that if g : X ! Y and f : Y ! Z are continuous, then
f g : X ! Z is continuous.
8. Consider the rational numbers Q as a subset of R, where the latter
has the usual metric topology.
(i) What is the closure of Q in R?
(ii) Show that every point in Q is open and closed in the subspace
topology.
9. Let X be a topological space, and A a subset of X with the subspace topology. Let i : A ! X be the inclusion map. Show that a map
f : Y ! A is continuous if and only if i f : Y ! X is continuous.
10. Suppose that X is a topological space with base B, and Y is a
subset of X. Show that the set
BY = {B \ Y | B 2 B}
is a base for the subspace topology on Y .
TOPOLOGY PROBLEMS
FEBRUARY 23—WEEK 2
1. Warmups
1. Let X and Y be topological spaces.
(i) Show that if X is discrete, every function f : X ! Y is continuous.
(ii) Show that if Y is indiscrete, every function f : X ! Y is continuous.
2. Proposition/Definition: For any set A, topological space X, and function
f : A ! X, there is a unique coarsest topology on A, the initial topology induced by X, such that f is continuous. The same is true given a family of spaces
Xi and maps f : A ! Xi .
Q
(i) Restate the definition of the product topology on a product i2I Xi in these
terms.
(ii) Why is it important that the product topology be as coarse as possible?
2. Products
3. Let X and Y be topological spaces. In each part below, determine whether the
statement is true or false. If true, prove it. If false, give a counterexample.
(i) If Y is discrete, and X is a subspace of Y , X must be discrete.
(ii) If X and Y are discrete spaces, X ⇥ Y must also be discrete.
(iii) If f : X ! Y is continuous and Y is discrete, X must be discrete.
4. Consider a family of topological spaces
Q Xi , i 2 I. Show that
Q if for each i 2 I,
Ci is a closed subset of Xi , then the set i2I Ci is closed in i2I Xi .
5. Let A be a subspace of X, and let B be a subspace of Y . Show that the product
topology on A ⇥ B is the same as the subspace topology on A ⇥ B inherited from
X ⇥Y.
6. We noted that the product and box topologies are the same for finite products.
Consider, instead,
Y
RN =
Xn
n2N
where Xn = R for all n, always with the metric topology. Define a map f : R !
RN , where for any t 2 R, f (t) = (t, t, t, . . . ).
(i) Show that f is continuous if RN has the product topology.
(ii) Show that f is not continuous if RN has the box topology. Hint: find an
element of the base for the box topology whose inverse image is not open.
7. In RN , consider the subset R1 consisting of all sequences (x0 , x1 , . . . , xn , . . . )
that are eventually zero: xn = 0 for all n larger than some N , which may depend
on the sequence.
(i) Find the closure of R1 in RN with the product topology.
(ii) Find the closure of R1 in RN with the box topology.
3. Separation Axioms
8. Using only finite spaces, find examples of the following:
(i) X is not T0.
(ii) X is T0, but not T1.
(ii) X is T1, but not T2.
8. Using only infinite spaces, find examples of the following:
(i) X is not T0.
(ii) X is T0, but not T1.
(ii) X is T1, but not T2.
TOPOLOGY PROBLEMS
MARCH 8—WEEK 3
1. Separation Axioms
1. Show that every closed subspace of a normal space is normal.
2. Let X be T 1, and let A be a subset of X. Then a point x 2 X is a limit
point of A (every neighborhood of x contains a point in A) if and only if every
neighborhood of x contains infinitely many points of A.
3. (i) Show that if X is regular, every pair of distinct points have neighborhoods
whose closures are disjoint.
(ii) Show that if X is normal, every pair of disjoint closed sets have neighborhoods
whose closures are disjoint.
4. Let p : X ! Y be a continuous surjection that is also closed (the image of any
closed set is closed). Show that if X is normal, then so is Y .
2. Compactness
5. Show that in the finite complement topology on R, every subspace is compact.
6. (i) Show that a finite union of compact subspaces of a space X must be compact.
(ii) Show that the infinite union of compact subspaces of a space X may not be
compact.
7. Show that if X is Hausdor↵, disjoint compact subspaces are contained in disjoint open neighborhoods.
8. Prove the following: Let f : X ! Y be a function, with Y compact Hausdor↵.
Then f is continuous if and only if the graph of f ,
f
is closed in X ⇥ Y .
= {(x, f (x)) | x 2 X},
3. Challenge problem
9. We say that a collection of subsets C of a space X have the finite intersection
property if for any finitely many C1 , C2 , . . . , Cn 2 C,
n
\
Cn 6= ;
i=1
Theorem: A space X is compact if and only if every collection of closed sets C in
X with the finite intersection property has a nonempty intersection, i.e.
\
6= ;.
C2C
TOPOLOGY PROBLEMS
MARCH 15—WEEK 4
1. Show that if Y is compact, the projection map p1 : X ⇥ Y ! X is closed; that
is, the image of any closed set is closed. (Hint: Tube Lemma.)
2. We say a space X is countably compact if every countable (spočetná) open
cover of X contains a finite subcover. Notice that every compact space is countably compact.
(i) Show that a space X is countably compact i↵ every nested sequence of nonempty
closed subsets C1 ◆ C2 ◆ · · · ◆ Cn ◆ . . . of X has nonempty intersection.
(ii) Challenge: find a topological space that is countably compact, but not compact.
3. We say a space X is limit point compact if every subset S ✓ X has a limit
point: some x 2 X such that every neighborhood of X contains a point of S (not
equal to x itself).
(i) Show that every compact space is limit point compact.
(ii) Consider the space X = Z+ ⇥ Y , where Z+ has the subspace topology from
R and Y is a two point space with the indiscrete topology. Show that X is limit
point compact, but not compact.
(iii) Show that a T 1 space is countably compact if and only if it is limit point
compact.
4. Let X be limit point compact.
(i) If Y is limit point compact as well, is X ⇥ Y ?
(ii) If f : X ! Y is continuous, is f (X) limit point compact?
(iii) If A is a closed subspace of X, is it limit point compact?
(iv) If X is a subspace of a Hausdor↵ space Y , is it closed?
5. Let X be an ordered set with the order topology: the base consists of all intervals (a, b), [a0 , b), and (a, b0 ], where a0 and b0 are the minimum and maximum
elements of X (if they exist).
Show that if every closed interval in X is compact, X has the least upper bound
property: if S ✓ X has an upper bound in X, then it has a least upper bound
(supremum) in X.
6. (Extreme Value Theorem) Let X be a topological space, and f : X ! R a
continuous function. Prove that f attains its maximum and minimum values on
any nonempty compact subspace of X.
TOPOLOGY PROBLEMS
MARCH 22—WEEK 5
1. Quotient Spaces and Maps
1. Show each of the following:
(i) Any quotient of a compact space is compact.
(ii) A quotient X/ ⇠ is T 1 if and only if the equivalence classes of ⇠ in X are closed.
(iii) If q : X ! X/ ⇠ is an open map, X/ ⇠ is Hausdor↵ if and only if the set
is closed in X ⇥ Y .
{(x, y) 2 X ⇥ Y | x ⇠ y}
2. (i) Let f : X ! Y be a continuous map. Show that if there is a continuous map
g : Y ! X so that f g is the identity map on Y (that is, if f has a section), then f
is a quotient map.
(ii) For any subspace A of a space X, a retraction of X onto A is a continuous
function r : X ! A such that r(a) = a for all a 2 A. Show that any retraction is a
quotient map.
2. Connected Spaces
3. Show that if X is an infinite set, it is connected in the finite complement topology.
4. Show that every discrete space is totally disconnected. Is every totally disconnected space discrete?
5. Show that any quotient of a connected space is connected.
6. Let ⇠ be an equivalence relation on a space X. Show that if each equivalence
class in X is connected and the quotient space X/ ⇠ is connected, then X is connected.
7. Let {Xi | i 2 I} be a family of connected spaces. Let X be the product space
Y
X=
Xi
i2I
and let ā = (ai )i2I be a point in X.
(i) For any finite subset F ✓ I, let XF denote the subspace of X consisting of all
points x̄ = (xi )i2I where xi = ai for all i 62 F . Show that XF is connected.
(ii) Show that the union Y of the spaces XF , where F ranges over all finite subsets
of I, is connected.
(iii) Show that X is the closure of Y , and therefore connected.
TOPOLOGY PROBLEMS
MARCH 29—WEEK 6
1. The “topologist’s sine curve” is the set
S = {(x, sin(1/x)) 2 R2 | x 2 (0, 1]}
while the “closed topologist’s sine curve” is
S̄ = {(x, sin(1/x)) 2 R2 | x 2 (0, 1]} [ ({0} ⇥ [ 1, 1])
(i) Show that S̄ is connected.
(ii) Show that S̄ is not path connected.
(iii) Show that S̄ is not locally connected.
2. Intermediate Value Theorem: Let f : X ! Y be continuous, X connected, and
Y an ordered set (with the order topology). If a, b 2 X, then for any r between
f (a) and f (b), there is c 2 X such that f (c) = r.
3. Is a product of path connected spaces path connected? If so, prove it. If not,
find a counterexample.
4. Let X be a space, and let ⇠ be the equivalence relation on X given by x ⇠ y if
and only if x and y belong to the same connected component of X. Show that the
quotient space X/ ⇠ is totally disconnected.
5. If a space X is locally connected, its connected components are clopen.
6. Let X be a space. Define x ⇠ y if there is no separation of X = A [ B into
disjoint open sets such that x 2 A and y 2 B.
(i) Show that ⇠ is an equivalence relation. Equivalence classes of ⇠ are called quasicomponents of X.
(ii) Show that the quasicomponent of a point x is equal to the intersection of all
clopen sets in X containing x.
(iii) Show that every component of X lies in a quasicomponent.
(iv) Show that if X is locally connected, its components and quasicomponents are
the same. Hint: Use #5 and parts (ii) and (iii) above.
7. Find a locally connected space that is not locally path connected.
8. Show that Q is not locally compact.
9. Show that if X is Hausdor↵ and locally compact at x 2 X, then for each
neighborhood U of x, there is a neighborhood V of x such that V̄ is compact, and
V̄ ✓ U .
TOPOLOGY PROBLEMS
APRIL 5TH—WEEK 7
1. Show that given homotopic maps h, h0 : X ! Y and homotopic maps k, k 0 : Y !
Z, the compositions k h and k 0 h0 are homotopic.
2. Given spaces X and Y , denote by [X, Y ] the set of homotopy classes of continuous maps from X to Y . Let I = [0, 1].
(i) Show that [X, I] contains a single element, for any space X.
(ii) Show that if Y is path connected, [I, Y ] contains a single element.
One way of defining contractibility of a space is this: A space X is contractible if
the identity map IdX : X ! X is nulhomotopic; that is, homotopic to a constant
map.
(i) Show that I and Rn are contractible.
(ii) Show that any contractible space is path connected.
(iii) Find a path connected space that is not contractible.
4. Show that if A is a convex subset of Rn (that is, for any a, b 2 A, the line segment
from a to b is contained in A), then any two paths f, g : I ! A between points
a, b 2 A are homotopic as paths.
5. Show that Rn \ {0} is homotopy equivalent to S n 1 . What happens if we take
Rn and remove two points? Or n points?
6. Let X be a path-connected space, and let a, b 2 X. Show that ⇡1 (X, a) '
⇡1 (X, b).
7. Let X be a space, A a subset of X. Recall that a retraction of X onto A is a
continuous map r : X ! A such that r(a) = a for all a 2 A. Show that if a 2 A,
then the map
r⇤ : ⇡1 (X, a) ! ⇡1 (A, a)
given by r⇤ ([f ]) = [r f ] is a surjective group homomorphism.
TOPOLOGY PROBLEMS
APRIL 12TH—WEEK 8
1. Given spaces Xi , i 2 I, we define a topology on the disjoint union
a
[
Xi = ({i} ⇥ Xi )
i2I
i2I
`
by taking the open sets to be those U ✓ i2I Xi such that U \ Xi is open for all
i 2 I. Show that this is indeed a topology on the disjoint union.
2. A map p : Y ! X is called a covering if for any x 2 X, there
` is a neighborhood
U of x such that for some I there is a homeomorphism : i2I U ! p 1 (U ) such
that p
is the identity on U .
Give, with proof, a covering of S 1 (by R, maybe).
3. For the following problems, you will need to consider the lifting of paths and
homotopies to covering spaces.
(i) Prove the following: If p : Y ! X is a covering and Y is simply connected, then
there is a bijection between ⇡(X, x) and p 1 (x).
(ii) Thinking of S 1 as the unit circle in C, use (i) and problem 2 to compute ⇡(S 1 , 1),
and give explicit representatives of each homotopy class.
(ii) Prove the following generalization of (i): For any covering p : Y ! X, any
x 2 X and y 2 p 1 (x), there is a bijection between p 1 (x) and the quotient group
⇡(X, x)/p⇤ ⇡(Y, y).
4. We will use algebraic topology to prove the Fundamental Theorem of Algebra:
Let f (x) = xn + c1 xn1 + + cn1 x + cn be a polynomial with n > 0 and each ci 2 C.
Then f has at least one root in C.
(i) Suppose f has no roots in C, and define
gr (s) =
f (re2⇡is )/f (r)
|f (re2⇡is )/f (r)|
Show that for all r 2 R, gr is a continuous map from I to S 1 , and that it forms a
loop with gr (0) = gr (1) = 1.
(ii) Show that any gr is homotopic to the constant
P loop at 1.
(iii) (Analysis, rather than topology) Fix R >
|ci | and R 1. Show that for all
x with |x| = R, the polynomial
ft (x) = xn + t(c1 xn
1
+ · · · + cn 1 x + cn )
has no roots on the circle of radius R for any 0  t  1.
(iv) Consider
ft (Re2⇡is )/ft (R)
gR,t (s) =
|ft (Re2⇡is )/ft (R)|
Show that this gives a homotopy from gR to !n , the loop that winds n times around
the origin.
(v) Conclude that n = 0. (How?)
5. (i) Show that a T 1 space X is completely regular if and only if it is homeomorphic to a subspace of a compact Hausdor↵ space Y , via some h : X ! Y .
(ii) In this case, let X be the closure of h(X) in Y . We call this the Stone-Čech
compactification of X. Show that X has the following universal property: for any
compact space K and embedding f : X ! K, there is a unique map f : X ! K
such that f h = f .
(iii) Show that is functorial from the category of completely regular (T 1) spaces
to the category of compact Hausdor↵ spaces.
TOPOLOGY PROBLEMS
APRIL 19TH—WEEK 9
1. Given a topological space X, we denote by C(X) the set of all continuous realvalued functions on X, and by C ⇤ (X) the set of all bounded real-valued continuous
functions. Prove the following:
(i) A topological space is completely regular i↵ it has the initial topology induced
by C(X), i.e. it has the coarsest topology in which every f 2 C(X) is continuous.
(ii) A topological space is completely regular i↵ it has the initial topology induced
by C ⇤ (X).
2. Prove a space X is completely regular i↵ the following equivalent conditions hold:
(i) Every closed subset C of X is an intersection of a family of zero sets of fi 2 C(X):
\
C = {x 2 A | fi (x) = 0}
i2I
(ii) X has a base consisting of the cozero sets of functions f 2 C(X), i.e. sets
{x 2 X | f (x) 6= 0} with f 2 C(X).
3. Give a direct proof of the Urysohn lemma for a metric space (X, d): for disjoint
closed A and B in X, consider
d(x, A)
f (x) =
.
d(x, A) + d(x, B)
4. Show that the Tietze extension theorem implies the Urysohn lemma.
5. We say that a space X is second countable if it has a countable base.
(i) Show that Rn in the metric topology is second countable.
(ii) The prime integer topology on Z>0 has a base consisting of sets
Up (b) = {b + np 2 Z>0 | n 2 Z}
with p prime and b 2 Z>0 . Show that this space is second countable and Hausdor↵,
but not compact.
6. Urysohn metrization theorem: Any second countable regular space is metrizable.
(i) Find a second countable Hausdor↵ space that is not metrizable.
(ii) Find a compact Hausdor↵ space which is not metrizable.
(iii) Show that a compact Hausdor↵ space is metrizable if and only if it is second
countable.
7. (i) Show that a topological space is locally compact Hausdor↵ if and only if
there is a compact Hausdor↵ space Y such that X is a subspace of Y with Y \ X
consisting of a single point.
(ii) Show that the space Y is unique up to homeomorphism fixing the subspace X.
8. (i) Show that the one point compactification of R is (homeomorphic to) S 1 .
(ii) Show that the one point compactification of Z>0 is (homeomorphic to) the subspace {0} [ {1/n | n 2 Z>0 } of R.
(iii) Find, with proof, the one point compactification of D2 .
TOPOLOGY PROBLEMS
APRIL 26TH—WEEK 10
1. (i) Show that the one point compactification of R is (homeomorphic to) S 1 .
(ii) Show that the one point compactification of Z>0 is (homeomorphic to) the subspace {0} [ {1/n | n 2 Z>0 } of R.
(iii) Find, with proof, the one point compactification of R2 .
2. Let X be a Hausdor↵ space. Then X is locally compact if and only if for every
x 2 X and neighborhood U of x, there is a neighborhood V of x with V̄ compact
and V̄ ✓ U .
3. Show that a continuous bijection f : X ! Y , with Y locally compact Hausdor↵,
is a homeomorphism if and only if it is proper, i.e. for any compact C in Y , f 1 (C)
is compact in X.
4. The Alexandrov extension of a space X, which we here denote by ↵X, is related
to one-point compactifications: the space ↵X has underlying set X [ {1}, with a
set U open in ↵X if and only if either U is open in X, or U contains 1 and U c is
compact in X.
(i) Show that for any space X, ↵X is compact.
(ii) Show that for any noncompact X, ↵X is a compact Hausdor↵ if and only if X
is locally compact Hausdor↵.
(iii) Show that ↵X is unique up to homeomorphism fixing X.
(iiv) Challenge question: Does ↵ give a functor from the category of topological
spaces with continuous maps to the category of compact spaces? From the category of locally compact Hausdor↵ spaces with continuous maps to the category of
compact Hausdor↵ spaces?
5. Prove the following:
(i) A space X has a compactification if and only if X is Tychono↵, i.e. completely
regular and Hausdor↵.
(ii) Theorem: Any locally compact Hausdor↵ space is Tychono↵.
6. Find a Tychono↵ space that is not locally compact. (Question: when does a
Tychono↵ space have a one-point compactification?)
7. Let X be Tychono↵. Recall that X denotes the Stone-Čech compactification
of X. Show that the following are equivalent:
(1) X has a unique (up to homeomorphism) compactification.
(2) X is compact, or | X \ X| = 1.
Hence Tychono↵ spaces with unique compactifications (sometimes called almost
compact spaces) admit one-point compactifications as well.
8. Show that any map f : B n ! S n
1
fixes a point on the boundary of B n .
Challenge Problem:
(i) Prove Helly’s Theorem: Let X1 , ..., Xn be a finite collection of convex subsets
of Rd , with n > d. If the intersection of any d + 1 of these sets is nonempty, then
the whole collection has a nonempty intersection. (Brouwer!)
(ii) Use (i) to prove the Centerpoint Theorem in two dimensions: For any set S of
points in R2 , there is a centerpoint xS such that for any line l through xS , there are
at least |S|/3 points of S on either side of l, where |S| denotes the size of S.
(iii) Generalize to higher dimensions. How does this relate to the one-dimensional
median of a subset S ⇢ R?
TOPOLOGY PROBLEMS
MAY 3RD—WEEK 11
1. A filter on a set X is a collection F of subsets of X satisfying
• If U 2 F and U ✓ V then V 2 F.
• If U, V 2 F then U \ V 2 F.
• ;2
/ F.
We say F is a proper filter if F ( P(X). We often think of a filter F on X as
containing “large” subsets of X: for U 2 F, statements of the form “For all x 2 U
...” mean something like “For most x 2 X...”
(i) Show that for any set X, the set of all cofinite subsets of X is a filter.
(ii) If you enjoy analysis, show that for any (complete) measure space (X, µ),
F = {A ✓ X | µ(Ac ) = 0}
is a filter.
(iii) Most importantly, show that for any topological space X and x 2 X, the set
Nx of neighborhoods of x is a filter on X.
2. We say that a filter F on a set X converges to a point x 2 X (and write F ! x)
if Nx ✓ F.
(i) Show that if X is Hausdor↵, any filter F on X converges to at most one point.
(ii) Let ā = (an )n2N be a sequence in a space X. By a tail of the sequence, we
mean a set tā (m) = {an : n m}. Show that the collection
Tail(ā) = {U ✓ X : 9m s.t. tā (m) ✓ U }
is a filter on X.
(iii) Show that if ā ! a then Tail(a) ! a. Thus filter convergence generalises
ordinary convergence.
3. An ultrafilter F on X is a maximal filter – one not contained in any larger proper
filter. Equivalently, a filter F is an ultrafilter if and only if for each subset U ✓ X
we have U 2 F or U c 2 F.
(i) Let x 2 X. Show that {U ✓ X : x 2 U } is an ultrafilter on X. Such ultrafilters
are called principal ultrafilters.
(ii) Let A be a nonempty subset of X, and let F = {U ✓ X | A ✓ U }. Show that
F is a filter, but that it is an ultrafilter if and only if A = {x} for some x 2 X.
Fact: Every filter is contained in an ultrafilter. (Zorn’s Lemma...)
(iii) Let X be the set of ultrafilters on X. Show that the sets {p 2 X : U 2 p}
(with U ranging over the subsets of X) form the base for a topology on X. Moreover, show that each element of the base is clopen. This is the Stone topology on
X.
(iv) Show that X, viewed as a discrete space, is a subspace of X, and that X is
compact Hausdor↵.
(v) Challenge: In fact, X is the Stone-Čech compactification of the discrete
space X. Check that X satisfies the universal property of the Stone-Čech compactification, and that X is dense in X.
4. Let X, Y be arbitrary topological spaces. The following results suggest that
filter convergence, while slightly less intuitive than convergence of sequences, gives
us equivalences where before we had only single implications.
(i) Let B be a nonempty subset of X. Then x 2 B if and only if there is an ultrafilter F on X such that B 2 F and F ! x.
(ii) A function f : X ! Y is continuous if and only if for every ultrafilter F on X,
if F ! x then f⇤ F ! f (x). Here f⇤ F = {V ✓ Y | f 1 (V ) 2 F}, which we call the
pushforward of F along f .
(iii) Show that a topological space X is compact if and only if every ultrafilter on
X is convergent.
5. Filters and Nets: Let X be a topological space, and let F be an ultrafilter on
X. We say that a net (xi )i2I is a section for F if for any F 2 F, there is a iF 2 I
such that xj 2 F for all j 2 I with j iF . Prove the following:
Theorem: For any space X, ultrafilter F on X, and section (xi )i2I of F, the
following are equivalent for any point x 2 X:
(1) The net (xi )i2I converges to x.
(2) The ultrafilter F converges to x.
6. Tychono↵ Theorem: It is possible to prove the Tychono↵ Theorem using
ultrafilters. Find a proof online and see if you can make sense of it.
TOPOLOGY PROBLEMS
MAY 10TH—WEEK 12
1. Recall that any metric space (X, d) gives rise to a uniform space (X, U ), where
U consists of the entourages U✏ = {(x, y) | d(x, y)  ✏} for each ✏ > 0.
Check that (X, U ) is in fact a uniform space.
2. Recall that a function f between uniform spaces (X, U ) and (Y, U ) is uniformly
continuous if for every entourage V 2 V, there is an entourage U 2 U such that for
any (x, y) 2 U , (f (x), f (y)) 2 V .
Show that if X and Y are metric spaces, a map f : X ! Y is uniformly continuous
by the usual ✏
definition if and only if it is uniformly continuous between the
associated uniform spaces.
3. Let (X, ⌧ ) be a compact Hausdor↵ space. Show that there is a unique uniformity
U on X such that the topology induced by U , ⌧ (U ), is ⌧ .
Hint: consider the collection of neighborhoods of the diagonal in X ⇥ X.
4. Cauchy Filters/Nets: Let (X, U ) be a uniform space.
(i) Let F be a filter on X. Define what it means for F to be Cauchy with respect
to the uniformity U .
(ii) Alternatively, let (xi )i2D be a net in X, i.e. a D-indexed collection of elements
of X. What should it mean for such a net to be Cauchy?
Definition: We say that a uniform space is complete if every Cauchy filter (or net)
converges.
(iii) Show that if X is a metric space, it is complete in the usual sense if and only
if its associated uniform space is complete in the sense of the definition above.
5. Hausdor↵ Completions: Show that for every uniform space X, there is a
Hausdor↵ uniform space Y and uniformly continuous map i : X ! Y with the
following universal property: for any uniformly continuous f : X ! Z, with Z a
Hausdor↵ uniform space, there is a unique uniformly continuous map g : Y ! Z
such that f = gi.
6. Compactness: We say that a uniform space (X, U ) is totally bounded if and
only if for every U 2 U there are x0 , . . . , xn 2 X such that X ✓ [ni=0 U [xi ]. Prove
the following:
Theorem: Let (X, U ) be a uniform space. Then (X, ⌧ (U )) is compact if and only
if (X, U ) is complete and totally bounded.