Download Topology (Maths 353). Problems

Topology (Maths 353). Problems
Theodore Voronov
January 27, 2003
Contents
1 Topological spaces and continuous maps
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2 Topological constructions
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3 Fundamental topological properties
4
4 Manifolds and surfaces
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5 Triangulations and Euler characteristic
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1
Topological spaces and continuous maps
Problem 1.1. Find all possible topologies on a two-point set: X = {a, b}.
Problem 1.2. Prove that open sets in Rn (in the usual sense) satisfy T1,
T2, T3, thus making a topology. Hint: the intersection of two open balls is
a union of open balls (check).
Problem 1.3. Check that a discrete topology (i.e., the collection of all subsets) is indeed a topology for an arbitrary set X.
Problem 1.4. Check that the collection {∅, X} for an arbitrary set X
(sometimes called the indiscrete topology) is indeed a topology.
Problem 1.5. For an arbitrary set X consider the collection F consisting of
the empty set ∅ and the complements of all finite subsets (i.e., X \ A where
A is finite). Prove that F is a topology, called the cofinite topology on X.
What is F is X is finite?
Problem 1.6. Sketch a nonempty open set in R considered with the cofinite
topology. Is it open in the usual sense? Does every subset in R open in the
usual topology belong to the cofinite topology? (If the answer is negative,
give a counterexample).
THEODORE VORONOV
Problem 1.7. Prove that the composition of continuous maps is continuous.
Problem 1.8. Let F denote the cofinite topology on R and E denote the
usual (Euclidean) topology. Is the “identity” map f : (R, F) → (R, E),
f : x 7→ x, a homeomorphism? Hint: check if both f and f −1 are continuous
directly from the definition.
Problem 1.9. (a) Prove that every two open finite intervals in R are homeomorphic. Hint: use parallel translations and dilations.
(b) Prove an open interval is homeomorphic to R. Hint: use the map
x 7→ tan πx, for x ∈ (−1, 1).
Problem 1.10. Let B n denote the open unit ball in Rn with center at the
origin, i.e., B n = {x ∈ Rn | |x| < 1}. Prove that the map f : Rn → B n ,
f : x 7→
x
∈ Rn
1 + |x|
is well-defined (i.e., that its image is indeed in B n ) and gives a homeomorphism B n ∼
= Rn .
Problem 1.11. Check that a stereographic projection establishes a homeomorphism S n \ {point} and Rn . Hint: write down the explicit formulae for
the projection and the inverse map.
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Topological constructions
Problem 2.1. Define R+ as a subspace of R consisting of all positive numbers. Show that R and R+ are homeomorphic. Hint: use the exponential.
Problem 2.2. Prove that the subspace of R3 specified by the conditions
x2 + y 2 − z 2 = −1, z > 0, is homeomorphic to R2 . Hint: make a sketch; you
might use a projection to establish a required homeomorphism.
Problem 2.3. Prove that the space SU (2) consisting of unitary 2 × 2matrices satisfying det A = 1 is homeomorphic to S 3 . Hint: write down
the equations specifying SU (2) as a subspace of all complex 2 × 2-matrices
explicitly.
Problem 2.4. Prove that for an identification space X/R, a map f : X/R →
Y is continuous if and only if f ◦ p : X → Y is continuous.
Problem 2.5. Let R be the following relation on R: xRy if and only if
x − y ∈ Z. Check that it is an equivalence relation.
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TOPOLOGY. Fall 2002. Problems
Problem 2.6. The identification space of R w.r.t. the equivalence relation
defined in the previous problem is denoted R/Z. Let p : R → R/Z be the
canonical projection. Show that p is an open map (i.e., the image of every
open set is open). Hint: consider a base of topology on R consisting of
intervals of length < 1/2 and show that p−1 (p(U )) is open for any such
interval U .
Problem 2.7. Check that the sets of the form p((a, b)) where |a − b| < 1/2
make a base of the identification topology for R/Z.
Problem 2.8. Consider a map f : R/Z → S 1 , where f : [x] 7→ e2πix . (The
circle is considered as a subspace of C: S 1 = {z ∈ C | |z| = 1}.)
(a) Check that f is well defined.
(b) Prove that f is continuous. Hint: use Problem 2.4.
z
(c) Check that g : S 1 → R/Z, g : z 7→ [ arg
] is the inverse map for f .
2π
(d) Show that g is continuous (hence f and g establish a homeomorphism
R/Z ∼
= S 1 ). Hint: use the base constructed in Problem 2.7.
Problem 2.9. Prove that a map f : Z → X × Y is continuous if and only
if the maps f1 = p1 ◦ f : Z → X and f2 = p2 ◦ f : Z → Y are continuous.
Problem 2.10. Show that R2 \ {(0, 0)} is homeomorphic to the product
space R × S 1 (infinite cylinder). Hint: use polar coordinates.
Problem 2.11. Show that the 2-torus defined as a surface of revolution in
R3 (a subspace of R3 ) is homeomorphic to each of the following spaces:
(a) the identification space of R2 w.r.t. the equivalence relation: (x, y) ∼
(x + 1, y) and (x, y) ∼ (x, y + 1);
(b) the identification space of I 2 (the unit square) w.r.t. the same equivalence
relation as above;
(c) the product space S 1 × S 1 .
Problem 2.12. Show that U (n) is homeomorphic to the product space S 1 ×
SU (n), where SU (n) is defined as the subspace consisting of the matrices
with unit determinant.
Problem 2.13. The real projective space RP n is defined as the identification
space of Rn+1 \{0} w.r.t. the following equivalence relation: v ∼ u if and only
if v = au, a 6= 0 (a nonzero real number). Check that RP n is homeomorphic
to an identification space of S n . What is the corresponding equivalence
relation on the sphere? Hint: use unit vectors.
Problem 2.14. The complex projective space CP n is defined similarly as
the identification space of Cn+1 \ {0} w.r.t. the equivalence relation v ∼ u if
and only if v = au, a 6= 0 (a nonzero complex number). Check that CP n is
homeomorphic to an identification space of S 2n+1 . What is the corresponding equivalence relation on the sphere?
P k kHint: use unit vectors w.r.t. the
Hermitian scalar product (u, v) =
u v̄ .
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THEODORE VORONOV
Problem 2.15. Show that RP 1 ∼
= S 1.
Problem 2.16. Show that CP 1 ∼
= S 2.
Problem 2.17. Show that RP n = Rn ∪ RP n−1 . More precisely, show that
in RP n there are subspaces homeomorphic to Rn and to RP n−1 , and that
the whole space is their union. (This gives an inductive description of the
structure of the projective space.) Hint: use coordinates in Rn+1 to specify
subspaces in RP n .
Problem 2.18. Prove a similar statement for CP n .
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Fundamental topological properties
Problem 3.1. Show that the following sets are closed:
(a) I = [0, 1] ⊂ R, (b) I n ⊂ Rn , (c) S n ⊂ Rn+1 , (d) the set of all noninvertible matrices in Mat(n), (e) the set of all orthogonal matrices in Mat(n),
(f) the set of all unitary matrices in Mat(n, C).
Problem 3.2. Show that the following spaces are Hausdorff:
(a) I n , (b) S n , (c) T 2 , (d) T n .
Problem 3.3. Show that the Klein bottle is Hausdorff. Hint: investigate
various “types” of points on I 2 (on the boundary, inside, etc.).
Problem 3.4. Show that an indiscrete space is non-Hausdorff.
Problem 3.5. Show that R with the cofinite topology is non-Hausdorff.
Problem 3.6. Prove that if both X and Y are Hausdorff then the product
space X × Y is Hausdorff.
Problem 3.7. Consider a topological space X. Suppose B is a base of the
topology of X. Show that if every cover of X by elements of B contains a
finite subcover then X is compact.
Problem 3.8. Show that if two spaces X and Y are homeomorphic and X
is compact then Y is also compact.
Problem 3.9. Prove that S n is compact for all n.
Problem 3.10. Can S n and Rn be homeomorphic? Justify your answer.
Problem 3.11. Prove that the closed n-disk Dn = {x ∈ Rn | |x| 6 1} is
compact for all n.
Problem 3.12. Is Z compact? Justify your answer.
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TOPOLOGY. Fall 2002. Problems
Problem 3.13. Show that the following spaces are compact:
(a) S n × I; (b) S n × S m ; (c) T n ; (d) T n × I; (e) RP n ; (f ) CP n .
Problem 3.14. Prove that the following topological groups are compact:
(a) SO(n); (b) O(n); (c) U (n); (d) SU (n).
Problem 3.15. Show that the topological group GL(n) is non-compact.
Hint: give an example of a matrix g ∈ GL(n) which is as far as you wish
from the zero matrix (look among the diagonal matrices).
Problem 3.16. Show that the topological group SL(n) is non-compact.
Hint: use the example from the previous problem modifying it if necessary.
Problem 3.17. Consider the straight line in Rn+1 (through the origin)
spanned by a nonzero vector v = (x1 , . . . , xn+1 ). Let Rv denote the reflection of the space Rn+1 in the plane orthogonal to v (i.e., the map which
fixes all the vectors in this plane and sends kv to −kv).
(a) Write down explicitly Rv (y) for an arbitrary vector y ∈ Rn+1 . Hint:
decompose y into the sum of a vector proportional to x and a vector in the
orthogonal plane.
(b) Write the operator Rv as a matrix.
(c) Check that [v] 7→ Rv is a well-defined map RP n → Mat(n + 1). Check
that it is injective.
(d) Using the homeomorphism theorem, show that the map above is a home2
omorphism of RP n with a subspace of Mat(n + 1) ∼
= R(n+1) .
Problem 3.18. Show that RP n is Hausdorff.
Problem 3.19. Construct an embedding of CP n into a Euclidean space.
Show that CP n is Hausdorff.
Problem 3.20. Suppose that topological spaces X and Y are homeomorphic.
(a) Prove that X is connected if and only if Y is connected.
(b) Prove that X is path-connected if and only if Y is path-connected.
Problem 3.21. Show that R is connected (you may assume without proof
that every finite segment [a, b] is connected).
Problem 3.22. Are the following spaces connected? path-connected?
(a) Z; (b) R; (c) O(2).
Problem 3.23. Show that the Klein bottle is connected.
Problem 3.24. Show that the torus T n is connected.
Problem 3.25. Use the notion of connectedness to prove that S 2 and T 2
are not homeomorphic.
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THEODORE VORONOV
Problem 3.26. Use the fact that for every matrix U ∈ U (n) there is an
invertible matrix g such that U = gDg −1 where D = diag(eix1 , . . . , eixn )
(here x1 , . . . , xn are real numbers) to show that the topological group U (n)
is connected. Hint: show that every point of U (n) can be joined by a path
with the identity matrix.
Problem 3.27. Use the fact that every matrix g ∈ GL(n, C) can be presented as a product g = RU where R is a Hermitian matrix with positive
eigenvalues and U is a unitary matrix to show that GL(n, C) is connected.
Hint: use the results of the previous problem; consider the matrix logarithm
to check that the space of Hermitian matrices with positive eigenvalues is
path-connected.
Problem 3.28. (a) Show that SO(2) is connected.
(b) Show that SO(3) is connected.
(c) Show that SO(n) is connected for all n.
Hint: for n > 3 use an argument similar to that in Problem 3.26.
Problem 3.29. A path-connected component of a topological space consists
of all points that can be joined by a path (i.e., two points belong to the same
component if they can be joined by a path, and to different components
otherwise).
(a) Prove that O(n) has exactly two path-connected components.
(a) Prove that GL(n) has exactly two path-connected components. Hint:
use an argument similar to that in Problem 3.27.
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Manifolds and surfaces
Problem 4.1. Consider the unit circle with center at the origin in R2 . Introduce two charts covering the circle using angles as coordinates (for one
chart you may take the circle without the point (1, 0) and count angles from
the positive direction of the x-axis; for another chart you may take the circle
without the point (−1, 0) and count angles from the negative direction of the
x-axis). Find the transition functions between these two charts.
Problem 4.2. Introduce two charts on S 1 using stereographic projections
and calculate the transition functions for them.
Problem 4.3. Find the transition functions for S n for the charts obtained
from stereographic projections with centers at the north pole and the south
pole.
Problem 4.4. Find the transition functions for RP 1 for the “canonical”
charts: (x1 : x2 ) = (y : 1) where x2 6= 0, and (x1 : x2 ) = (1 : y 0 ) where
x1 6= 0.
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TOPOLOGY. Fall 2002. Problems
Problem 4.5. Do the same for CP 1 . Write the answer in the complex and
the real forms.
Problem 4.6. Do the same for RP n and CP n . (You have to introduce some
notation to number charts and coordinates in them. It is suggested that the
i
coordinates in the j-th chart will be denoted y(j)
, with i = 1, . . . , j − 1, j +
j
1, . . . , n + 1, and with a “fake coordinate” y(j) = 1 (constant).)
Problem 4.7. Use coordinates to establish homeomorphisms between S 1
and RP 1 , and between S 2 and CP 1 .
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Triangulations and Euler characteristic
Problem 5.1. Using suitable triangulations find the Euler characteristics
for the following topological spaces:
(a) I = [0, 1]
Answer: 1
2
(b) I
Answer: 1
1
(c) S
Answer: 0
(d) S 2
Answer: 2
1
(e) S × I
Answer: 0
(f ) closed Möbius strip M
Answer: 0
Problem 5.2. Consider two different triangulations for each of the given
topological spaces and verify that the Euler characteristic does not depend
on a choice of triangulation: (a) I 2 ; (b) S 1 ; (c) S 2 .
Problem 5.3. (a) For a square check that the number of vertices (4) minus
the number of edges (4) plus the number of faces (1) coincides with the Euler
characteristic (1).
(b) Generalize to show that the Euler characteristic of a surface can be
calculated from any “tiling” by (homeomorphic images of) squares, each two
of them either not meeting or meeting by a single common edge.
Calculate the Euler characteristic of S 2 considering it as the boundary of a
cube and using tiling by squares.
Problem 5.4. Triangulate the following spaces and find the Euler characteristic:
(a) Klein bottle K
Answer: 0
(b) RP 2
Answer: 1
2
(c) T
Answer: 0
Hint: triangulate the square using the subdivision into 9 smaller squares by
lines parallel to the sides. Triangulating surfaces, mark vertices carefully,
keeping track of all identifications.
Problem 5.5. Using the excision formula or triangulations show that the
Euler characteristic of the sphere with k holes equals 2 − k.
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THEODORE VORONOV
Problem 5.6. (a) Use the excision formula to prove for the standard surface
Hg2 (the sphere with g handles) that χ(Hg2 ) = 2 − 2g.
(b) Use the excision formula to prove for the standard non-orientable surface
Mν2 (the sphere with ν Möbius strips) that χ(Mν2 ) = 2 − ν.
Hint: use the results of Problem 5.1 and Problem 5.5.
Problem 5.7. A surface Xg,ν is obtained from the sphere by attaching g
handles and gluing in ν Möbius strips. Find its Euler characteristic.
Problem 5.8. Prove that the surface Xg,ν defined in Problem 5.7 is homeomorphic to the standard surface M2g+ν . Hint: apply the classification theorem for closed surfaces and the result of the previous problem.
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