Download Math 201 Topology I

Math 201
Topology I
Exercises of Dr. Hicham Gebran
[email protected]
Lebanese University, Fanar, Fall 2013-2014
1
Metric spaces and topological spaces
1. The purpose of this exercise is to prove the fundamental theorem that any nontrivial interval
of IR contains rational and irrational numbers. Let a and b be arbitrary real numbers such
that a < b.
a) Let q be an integer such that q >
p > qa. Show that pq belongs to ]a, b[.
1
b−a
and let p be the smallest integer that satisfies
√
2
b) Let q be an integer such that q > b−a
and let p be the smallest integer that satisfies
√
qa
p
√
p > 2 . Show that q 2 belongs to ]a, b[.
c) Conclude.
2. Let d be a metric on a set X. Show that |d(x, y) − d(x, z)| ≤ d(y, z). This is the second form
of the triangle inequality.
S
T
B(a, r).
B(a, r) and
3. Let (X, d) be a metric space and let a ∈ X. Find
r>0
r>0
4. Let A and B be two subsets of a metric space. Prove the following.
◦
◦
a) A ⊂ B ⇒ A ⊂ B.
◦
◦
◦
◦
◦
◦
b) A ∩ B = A ∩ B.
c) A ∪ B ⊂ A ∪ B and the inclusion may be strict.
d) A ⊂ B ⇒ Ā ⊂ B̄.
e) A ∪ B = Ā ∪ B̄.
f) A ∩ B ⊂ Ā ∩ B̄ and the inclusion may be strict.
1 1 5. Show that
n ∈ IN∗ is not closed but that
n ∈ IN∗ ∪ {0} is closed.
n
n
6. Show that the set of integers is closed in the usual topology of IR.
7. a) Let A ⊂ IR be bounded from above. Prove that sup A ∈ Ā.
b) Let A ⊂ IR be bounded from below. Prove that inf A ∈ Ā.
1
◦
8. a) Prove that {A = {A .
b) Deduce that {Ā = int{A .
9. Let A be a nonempty set of a metric space X. Define the distance from a point x ∈ X to
the set A by
dist (x, A) = inf dist (x, y).
y∈A
a) Prove that x ∈ Ā if and only if dist (x, A) = 0.
b) Show that |dist (x, A) − dist (y, A)| ≤ dist (x, y) for all x, y ∈ X.
c) Prove that A ⊂ B ⇒ dist (x, B) ≤ dist (x, A).
d) Show that dist (x, A) = dist (x, Ā).
e) Prove that the following properties are equivalent
(i) ∀ x ∈ IRN , dist (x, A) = dist (x, B).
(ii) Ā = B̄.
10. Let X be a metric space. For an nonempty subset A ⊂ X, we set
diam(A) = sup{d(x, y)|x, y ∈ A}.
a) Find diam ]0,1[.
b) Show that if A ⊂ B then diam(A) ≤ diam(B).
c) Show that diam(A) = diam(Ā).
d) Show that diam(B(a, r)) ≤ 2r and diam(B 0 (a, r)) ≤ 2r. Show that these inequalities may
be strict.
e) A subset A ⊂ X is called bounded if diam(A) < ∞. Show that A is bounded if and only
if it is contained in a ball.
f) Show that a subset A ⊂ IR is bounded in the usual distance if and only if it is bounded
from above and from below.
11. Let A be a subset of metric space X and let x be a limit point of A. Show that any
neighborhood of x contains infinitely many points of A. Deduce that a finite set has no limit
points.
12. Let (X, d) be a metric space and A ⊂ X.
a) Prove that A is closed if and only it contains its boundary.
b) Prove that A is open if and only it does not meet its boundary.
13. Let (Aλ )λ∈L be an arbitrary family of subsets of a metric space. Show that
[
A¯λ ⊂
λ∈L
[
λ∈L
Give an example where equality fails.
2
Aλ .
14. Let (Aλ )λ∈L be an arbitrary family of subsets of a metric space. Show that
\◦
Aλ ⊂
λ∈L
\
◦
Aλ .
λ∈L
Give an example where equality fails.
◦
◦
√ √
15. Let A = Q∩] 2, 3] and B = (IR\Q) ∩ [0, 1[. Find A, Ā, B and B̄.
√
16. Which one of the following sets is dense in IR? Q ∪ { 2}, Q\{1}, Q\Z.
17. The purpose of the exercise is to show that the set D = 2mn , m ∈ Z, n ∈ IN is dense in IR.
Let x < y be two real numbers. Set = y − x. Choose an integer n such that n > 1. Find
an integer m such that x < m2−n < y. Conclude.
18. Let r > 0 and let ||x|| be a norm on IRN . Prove the following.
a) The closure of the open ball B(a, r) = {y ∈ RN | kx − ak < r} is the closed ball B 0 (a, r) =
{x ∈ IRN | kx − ak ≤ r}.
b) The interior of the closed ball is the corresponding open ball.
c) The boundary of the open ball B(a, r) as well as the boundary of the closed ball B 0 (a, r)
is the sphere S(a, r) = {x ∈ IRN | kx − ak = r}.
d) Show that the above relations do not hold in an arbitrary metric space.
19. Let A and B be two subsets of a metric space X.
a) Show that if A is open then A ∩ B ⊂ A ∩ B. Give an example where the inclusion is
strict.
b) Suppose that A and B are open. Prove that the following conditions are equivalent.
(i) A ∩ B = ∅.
(ii) A ∩ B = ∅.
(iii) A ∩ B = ∅.
20. Let (X, d) be a metric space.
a) Show that ∂(A ∪ B) ⊂ ∂A ∪ ∂B for any subsets A and B of X.
b) Show that the inclusion above may strict.
c) Show that equality holds if Ā ∩ B = A ∩ B̄ = ∅. Hint. It may be useful to establish that
under this condition we have (A ∪ B)◦ = A◦ ∪ B ◦ .
21. Let X be a set and let A and B be proper subsets of X. Consider the collection T =
{∅, A, B, X}. Under which conditions on A and B is T a topology?
22. Let X be an infinite set equipped with the finite complement topology. Show that every
nonempty open set is dense in X.
23. Let X = {a, b, c, d, e} and consider the topology
T = {∅, X, {a}, {a, b}, {a, c}, {a, c, d}, {a, b, c, d}, {a, b, c}}.
a) Find the closed subsets of X.
b) Find the closure of {a}, {b} and {c, e}.
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c) Find all the neighborhoods of c.
◦
d) Let A = {a, b, c}. Find A, Ā and ∂A.
e) Is (X, T ) a Hausdorff space?
24. For every n ∈ IN, set En = {k ∈ IN|k ≥ n} = {n, n + 1, n + 2, . . .} and let
T = {∅} ∪ {En , n ∈ IN}.
a) Show that T is a topology on IN.
b) Find the open sets containing 6.
c) Determine the closed sets of IN .
d) Find the closure of {7, 24, 47, 85} and {3, 6, 9, 12, . . .}.
e) Determine the dense subsets of IN.
25. Let X be a topological space and let A ⊂ X. Show that if A is closed or open then ∂A has
an empty interior.
26. a) Let A be a subset of a Hausdorff space X and let x be a limit point of A. Show that
every neighborhood of x contains infinitely many points of A.
b) Does the above result hold in an arbitrary topological space X?
c) Let X be a topological space, A ⊂ X and let x be a limit point of A. Is x a limit point
of Ā? Show that if A has no isolated points, then Ā also has no isolated points.
27. a) If (Tα ) is a collection of topologies on a set X, show that ∩Tα is a topology on X. Is ∪Tα
a topology on X?
b) Let (Tα ) be a collection of topologies on X. Sow that there is a unique smallest topology
on X containing all Tα , and a unique largest topology contained in all Tα .
c) Let X = {a, b, c} and let
T1 = {∅, X, {a}, {a, b}}
and
T2 = {∅, X, {a}, {b, c}}.
Find the smallest topology containing T1 and T2 and the largest topology contained in T1
and T2 .
28. Show that if B is a basis for a topology T on a set X, then T equals the intersection of all
topologies on X that contain B.
29. Let B be the collection of all intervals of the form ]a, b[ where a, b ∈ Q. Show that B is a
basis for the usual (or standard) topology of IR.
30. Let X be a set equipped with the finite complement topology and let Y ⊂ X. Show that
the subspace topology of Y coincides with the finite complement topology on Y .
31. Let X be a topological space, F be a subspace of X and A ⊂ F .
a) Let AF denote the closure of A in F . Show that AF = Ā ∩ F where Ā is the closure of
A in X.
◦
◦
◦
b) Let AF denotes the interior of A with respect to F . Show that A ⊂ AF . Give an example
where equality fails.
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32. Let X and Y be two topological space. Let A ⊂ X and B ⊂ Y . Show that A × B = Ā × B̄.
33. a) Show that a subspace of a Hausdorff space is a Hausdorff space.
b) Show that the product of two Hausdorff spaces is a Hausdorff space.
34. Show that X is a Hausdorff space if and only if the diagonal ∆ = {(x, x)|x ∈ X} is closed
in X × X.
35. Let (X1 , d1 ), . . . , (Xn , dn ) be n metric spaces. Let X = X1 × · · · × Xn . For x = (x1 , . . . , xn )
and y = (y1 , . . . , yn ), define
ρ1 (x, y) =
n
X
di (xi , yi )
i=1
ρ2 (x, y) =
n
X
!1/2
2
di (xi , yi )
i=1
ρ∞ (x, y) = max di (xi , yi ).
i=1...,n
Show that ρ1 , ρ2 and ρ∞ are metrics on X that generate the product topology on X.
2
Continuous functions and homeomorphisms
1. Let f : X → Y be a function between two topological spaces. Show that the following
conditions are equivalent.
(i) f is continuous.
(ii) f (Ā) ⊂ f (A) for any subset A ⊂ X.
(iii) f −1 (B) ⊂ f −1 (B) for any subset B ⊂ Y .
(iv) f −1 (int B) ⊂ int f −1 (B) for any subset B ⊂ Y .
Give an example where the inclusion in (ii) is strict.
2. Let X be a set and A ⊂ X. The characteristic function of A denoted by 1A is defined by
(
1
if x ∈ A
1A (x) =
0
if x ∈
/ A.
Let now X be a topological space. Show that the set of discontinuity points of 1A is precisely
∂A. Conclude that 1A is continuous if and only if A is clopen.
3. Let X be a topological space and let f : X → IR be continuous at a point x0 . Show that if
f (x0 ) > 0 then there exists a neighborhood U of x0 such that f (x) > 0 for all x ∈ U .
4. Let X be a topological space, Y be a Hausdorff space and f : X → Y be continuous. Show
that the graph of f , G(f ) := {(x, f (x)) | x ∈ X} is closed in X × Y .
5. Let X and Y be two topological spaces and let π : X × Y → X be the projection onto X.
Show that π is an open map i.e. that π maps open sets into open sets. Show that π need
not be a closed map i.e. does not map closed sets into closed sets.
6. Let f : X → Y be a function between two S
topological spaces. Suppose that there is a family
of open set (Uα )α∈A in X such that X = α∈A Uα . Show that f |Uα is continuous for every
α ∈ A if and only if f : X → Y is continuous.
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7. (The pasting lemma). Let X and Y be topological spaces such that X = A ∪ B where
A and B are closed in X. Let f : A → Y and g : B → Y be continuous. Show that if
f (x) = g(x) for all x ∈ A ∩ B, then the function h : X → Y defined by h(x) = f (x) for
x ∈ A and h(x) = g(x) for x ∈ B is continuous.
8. Let X be a topological space and let Y be a Hausdorff space. Let f : X → Y and g : X → Y
be two continuous functions.
a) Show that {x ∈ X|f (x) = g(x)} is closed in X.
b) Let A ⊂ X. Show that if f and g coincide on A then they coincide on Ā. In particular,
if two continuous functions coincide on a dense subset, then they coincide everywhere.
c) Suppose that X is metrizable and Y is arbitrary. Show that the conclusions of the
preceding questions remain true.
9. Let (X, d) be a metric space.
a) Show that every closed subset of X is a countable intersection of open subsets of X.
b) Deduce that every open subset of X is a countable union of closed subsets of X.
10. (Every metric space is normal). Let (X, d) be a metric space and let A and B be two
closed disjoint subsets of X. Show that there exists two open disjoint subsets U and V such
that A ⊂ U and B ⊂ V . Otherwise stated, in a metric space we can separate two closed
disjoint sets by two open sets. Hint. Let U = {x ∈ X | d(x, A) < d(x, B)}.
11. (Urysohn lemma in metric spaces). Let A and B be two closed disjoint subsets of
a metric space (X, d). Show that there is continuous function f : X → [0, 1] such that
f (A) = {0} and f (B) = {1}. Otherwise stated, we can separate two closed disjoint subsets
d(x, A)
by a continuous function. Hint. Consider the function f (x) =
.
d(x, A) + d(x, B)
12. Let IR2 be equipped with the usual topology. Which of the following subsets of IR2 is closed?
open? neither? Justify your answer.
a) {(x, y) | y > x2 }.
d) IR2 \{(0, 0)}.
b) {(x, y) | y ≤ x2 }.
c) {(x, y) | 0 < x2 + y 2 < 1}.
e) {(x, y) | 1 ≤ x2 + y 2 ≤ 2}.
13. Let f : (X, d1 ) → (Y, d2 ) be a function between two metric spaces. We set
ρ(x, y) = d2 (f (x), f (y)).
a) Show that ρ is a distance on X if and only if f is one to one.
b) Suppose that f is continuous and one to one. Show that the topology generated by d1 is
finer than the topology generated by ρ. Hint. Use the continuity of f to prove that
∀ x ∈ X, ∀r > 0, ∃ δ > 0 such that Bd1 (x, δ) ⊂ Bρ (x, r).
c) Show that if f is a homeomorphism, then d1 and ρ are topologically equivalent i.e. they
generate the same topology on X.
14. Let f : X → Y be a homeomorphism between two topological spaces. Prove that for any
subset A ⊂ X the following hold.
(i) f (A) = f (A).
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(ii) f (A◦ ) = f (A)◦ .
(iii) f (∂A) = ∂f (A).
Remark. (ii) is deducible from (i), and (iii) is deducible from (i) and (ii).
15. Show that in IR3 , a sphere, an ellipsoid and the boundary of a cube are homeomorphic.
16. Let f : IR → IR be strictly monotone and surjective. Show that f is a homeomorphism.
17. Let Mn (IR) denote the set of n × n real matrices .
a) For A = (aij ) ∈ Mn (IR), set ||A|| = max{|aij |, i, j = 1, . . . n}. Show that || · || is a norm.
Thus by setting d(A, B) = ||A − B||, we get a distance on Mn (IR).
2
b) Show that equipped with the above distance Mn (IR) is homeomorphic with IRn .
c) Show that the function det : Mn (IR) → IR is continuous. Deduce that the set GL(n, IR)
of invertible matrices of Mn (IR) is open. Deduce also that the set of matrices with positive
determinant is open in Mn (IR).
d) Can you prove that GL(n, IR) is dense in Mn (IR)?
3
Compactness
1. a) Let T and T 0 be two topologies on a set X. Suppose that T 0 is finer that T . Show that
(X, T 0 ) has less compact subsets than (X, T ).
b) Show that if (X, T ) and (X, T 0 ) are both compact Hausdorff, then either T and T 0 are
equal or they are not comparable.
2. Show that a finite union of compact subspaces of a space X is compact.
3. Let Kn be a decreasing sequence of compact sets in a Hausdorff space X. Let K = ∩Kn .
Show that if U is an open set containing K, then U contains Kn for all n large enough.
4. Let A and B be two disjoint compact subspaces of a Hausdorff space X. Show that there
exist two disjoint open sets U and V containing A and B respectively.
5. Let X and Y be two two topological spaces. Show that if Y is a compact, then the projection
π1 : X → Y → X is a closed map. Hint. Let A be a closed set of X × Y . Use the tube
lemma to show that X − π1 (A) is open.
6. Let f : X → Y where Y is compact space. Show that if the graph of f , G = {(x, f (x))|x ∈
X} is closed in X × Y , then f is continuous. Hint. Show that the inverse image under f
of a closed set is closed by using the previous exercise.
7. Let A, B ⊂ IRn . We set A + B = {x + y|x ∈ A, y ∈ B}.
a) Show that if A and B are compact, then A + B is compact.
b) Show that if A is compact and B is closed then A + B is closed.
8. Let X be a compact metric space. Let (xn ) ⊂ X be a sequence with a unique cluster point
a. Show that the sequence (xn ) converges to a.
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9. Let X be a metric space. For A, B ⊂ X, A, B 6= ∅, we set
d(A, B) = inf d(x, B) = inf inf d(x, B).
x∈A
x∈A y∈B
a) Show that if A is compact and B is closed with A ∩ B = ∅, then d(A, B) > 0. Does the
result hold if A is not assumed compact?
S
b) Show that {x ∈ X|d(x, A) < r} = a∈A B(a, r).
c) Suppose that A is compact and U is an open set containing A. Show that there exists
> 0 such that {x ∈ X|d(x, A) < } ⊂ U . Does the result hold if A is not assumed compact?
10. Show that the functions x 7→ x2 and x 7→ sin(x2 ) are not uniformly continuous on IR.
11. Let X be a locally compact metric space and let K ⊂ X be compact. Then there exists
r > 0 such that {x ∈ X|d(x, K) ≤ r} is compact. Hint.
Show first that for each
x ∈ X, there exists rx > 0 such that B 0 (x, rx ) is compact. Then consider the open covering
(B(x, rx /2))x∈K of K
12. a) Is Q locally compact?
b) Show that a closed subspace of a locally compact space is locally compact.
13. Prove that two homeomorphic locally compact Hausdorff spaces have homeomorphic one
point compactification.
14. Show that the one point compactification of IN∗ is homeomorphic with the subspace { n1 , n ∈
IN∗ } ∪ {0} of IR.
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Connectedness
1. Let T and T 0 be two topologies on a set X. Suppose that T 0 is finer that T . Show that
(X, T 0 ) has less connected subsets than (X, T ).
2. Let (An ) be a sequence
S of connected subspaces of a topological space X, such that An ∩
An+1 6= ∅. Show that An is connected
3. Let X be a topological space. Let (Aλ )λ∈L be a collection of connected subspaces of X.
S Let
A be a connected subspace of X such that A ∩ Aλ 6= ∅ for all λ ∈ L. Show that A ∪ ( Aλ )
is connected.
4. Show that if X is an infinite set, it is connected in the finite complement topology.
5. Let X be a topological space and A ⊂ X. Show that if C is a connected subspace of X that
meets both A and X − A, then C meets the boundary of A.
6. (a) Show that no two of the spaces ]0,1[, ]0,1] and [0,1] are homeomorphic. Hint.
happens if you remove a point from each of these spaces?
What
(b) Show that IRn and are not homeomorphic if n > 1
(c) Suppose that there exists an imbedding f : X → Y and an imbedding g : Y → X. Show
that X and Y need not be homeomorphic.
7. Let S 1 denote the unit circle in IR2 and let f : S 1 → IR be continuous. Show that there
exists a point x ∈ S 1 such that f (x) = f (−x).
8
8. Let f : [0, 1] → [0, 1] be continuous. Show that there exists x ∈ [0, 1] such that f (x) = x.
We say that f has a fixed point.
9. (a) Is the product of path connected sets path connected?
(b) If A is path connected, is Ā path connected?
(c) If (Aλ ) isSa collection of path connected subspaces of a topological space such that
T
Aλ 6= ∅, is Aλ path connected?
(d) Is the continuous image of a path connected space path connected?
◦
10. If A is connected does it follow that A and ∂A are connected?
◦
11. Let X be a topological space and let A ⊂ X. We say that A is nowhere dense if Ā = ∅. We
say that A is totally disconnected if the only connected subsets of A are the singletons.
a) Give an example of a nowhere dense set and an example of a totally disconnected set.
b) Give an example of a totally disconnected set which is not nowhere dense. Give an
example of a nowhere dense set which is not totally disconnected.
c) Let A ⊂ IR be closed. Show that the following statements are equivalent.
(i) A is totally disconnected.
(ii) A is nowhere dense.
12. Let X be a Hausdorff space. Let T
(Kn ) be a decreasing collection of compact and connected
subspaces of X. Show that K = Kn is connected. Hint.T If C ∪ D is a separation of K,
separate C and by two open sets U and V and show that (Kn − (U ∪ V )) is not empty.
13. Let X be a connected and locally connected space. Let Ω be a proper open subset of X and
let C be a component of Ω. Show that C̄ ∩ ∂Ω 6= ∅. Reason by contradiction.
14. Let X be a locally connected space. Let Ω ⊂ X and let C be a connected component of Ω.
Show that ∂C ⊂ ∂Ω. Give another proof of the previous exercise.
5
Complete metric spaces
1. (a) Let d(x, y) = |x3 − y 3 | for x, y ∈ IR. Show that (IR, d) is complete.
(b) Let ρ(x, y) = |ex − ey | for x, y ∈ IR. Show that (IR, ρ) is not complete.
2. Let C([a, b]) denote the space of continuous functions f : [a, b] → IR. We equip C([a, b]) with
the distance d(f, g) = supx∈[a,b] |f (x) − g(x)|. Show that (C([a, b]), d) is complete.
Z 1
3. We equip C([0, 1]) with the distance d1 (f, g) =
|f (x) − g(x)| dx. The purpose of this
0
exercise is to show that (C([0, 1]), d1 ) is not complete.
(a) Define a sequence of functions (fn )n≥2 as follows:


if 0 ≤ x ≤ 21
1
fn (x) = −nt + n2 + 1 if 12 ≤ x ≤ 12 + n1


0
if 12 + n1 ≤ x ≤ 1.
Draw the graph of (fn ) and convince yourself that (fn ) is continuous.
9
(b) Show that if m ≥ n, then d1 (fn , fm ) =
in (C([0, 1]), d1 ).
1
1
2n − 2m .
Conclude that (fn ) is a Cauchy sequence
Z
(c) Suppose that (fn ) converges to f in (C([0, 1]), d1 ). Show that
Z 1
|f (x)| dx = 0.
and
1/2
|f (x) − 1| dx = 0
0
1/2
(d) Conclude.
4. We equip IR2 with the distance ρ1 ((x, y), (x0 , y 0 )) = |x − x0 | + |y − y 0 |. Let f : IR2 → IR2 be
defined by
2
1
1
3
f (x, y) = ( sin x + cos y, cos x + sin y).
5
5
5
5
(a) Show that f is a contraction.
(b) Deduce that the system of equations
(
2 sin x + cos y = 5x
cos x + 3 sin y = 5y
has a unique solution.
5. Let X =]0, +∞[. For x, y ∈ X, we set d(x, y) = | ln x − ln y|.
(a) Show that d is a distance on X.
(b) Show that (X, d) is complete.
(c) Let f : X → X be a differentiable function such that |f 0 (x)| ≤ k|f (x)| for all x ∈ X
and some k ∈ [0, 1[. Give an example of such function. Show that f has a unique fixed
point.
6. Let X be a metric space.
a) Suppose that for some ε > 0, every ε−ball in X has compact closure. Show that X is
complete.
b) Suppose that for each x ∈ X there exists ε > 0 such that the ball B(x, ε) has compact
closure. Show by means of an example that X need not be complete.
7. Let (X, dX ) and (Y, dY ) be metric spaces with Y complete. Let A ⊂ X. Show that if f :
A → Y is uniformly continuous, then f can be uniquely extended to a uniformly continuous
function g : Ā → Y .
8. Two metrics on a set X are said to be metrically equivalent if the identity maps i : (X, d) →
(X, d0 ) and its inverse are both uniformly continuous.
¯ y) = min(d(x, y), 1) then d and d¯ are metrically
a) Show that if d is metric on X and d(x,
equivalent.
b) Show that if d and d0 are metrically equivalent, then (X, d) is complete if and only if
(X, d0 ) is complete.
9. Show that a metric space (X, d) is complete if and only if for every nested sequence A1 ⊃
A2 ⊃ · · · of nonempty closed subsets of X such that diamAn → 0, the intersection ∩An is a
singleton.
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