1 Week 1, Prof. Archana S. Morye - ncmath.org · 1 C A : This m n matrix is called the Jacobian of...

Annual Foundation School ICEMS, Kumaun University, Almora

1–27 December, 2012

Topology: Calculus of Several VariablesAssignment

December 1, 2014

Notations and Definitions

• Let ej , for 1 ≤ j ≤ n denotes the vector in Rn whose jth co-ordinate is 1 and whose other co-ordinates areall 0. We shall call {e1, . . . , en} the standard basis of Rn.

• For x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ Rn, |x| denotes the norm of x and is given by√x2

1 + · · ·+ x2n, and

x · y = x1y1 + · · ·+ xnyn denotes dot product of x and y.

• L(X,Y ) denotes the set of all linear transformations of the vector space X into the vector space Y . Insteadof L(X,X), we shall simply write L(X).

• For A ∈ L(Rn,Rm), we shall define the norm ||A|| of A to be the sup of all numbers |Ax|, where x rangesover all vectors in Rn with |x| ≤ 1.

• Sn denotes the unit circle in Rn+1 with center at origin, i.e., {x ∈ Rn+1 | |x| = 1}• We will denote (0, . . . , 0) ∈ Rn by 0, for n ≥ 2.

• Let f be a differentiable function from E ⊂ Rn into R. Then, we will denote by the gradient of f atx = (x1, . . . , xn) ∈ E, by a vector (∇f)(x) =

∑ni=1( ∂f∂xi

)(x)ei.

• Dual of a real vector space V is defined by L(V,R) and is denoted by V ∗.

• If f : U → Rm is differentiable at x ∈ U ⊂ Rn, then we will denote by dfx : Rn → Rm the differential of fat point x (or sometimes we will use dfx for the same).

• If f : U → Rm is a differentiable map at x ∈ U ⊂ R, then dfx is a linear map from R to Rm, thereforedfx(t) = tdfx(1), for all t ∈ R. If f is differentiable at every point of U , then we call a function f ′ : U → Rm

given by f ′(x) = dfx(1) the derivative of f .

• Let U ⊂ Rn be an open set, and let f : U → R. Let x0 ∈ U . We say that the ith-partial derivative of f

exists at x0 for i = 1, . . . , n, if limt→0

f(x0 + tei) + f(x0)

texists. In that case, we denote the above limit by

∂f∂xi

(x0). If ∂f∂xi

(x) exists for all x ∈ U , then we will denote by ∂f∂xi

: U → R the realvalued function on U ,

x 7→ ∂f∂xi

(x).

• Let U ⊂ Rn be an open set, and let f = (f1, . . . , fn) : U → Rm be a map that is differentiable at x0. Thenthe matrix of the linear map dfx0 : Rn → Rm, with respect to the canonical basis of Rn and Rm, equals

∂f1∂x1

(x0) · · · ∂f1∂xn

(x0)...

. . ....

∂fm∂x1

(x0) · · · ∂fm∂xn

(x0)

.

This m× n matrix is called the Jacobian of f at x0, and is denoted by Jfx0of Jf(x0).

• A set of the form W = {(x1, . . . , xn) |αi < xi < βi, 1 ≤ i ≤ n} ⊂ Rn, where αi ≤ βi ∈ R for all 1 ≤ i ≤ nis called an open rectangle in Rn, and its volume is defined as Vol(W ) =

∏ni=1(βi − αi).

• A subset F of Rn is said to be set of measure zero if for given ε > 0 there exists a countable open cover{Wα}α∈Λ of F , i.e., F ⊂ ⋃α∈ΛWα, where Wα is an open rectangle for every α ∈ Λ, and

∑α∈Λ Vol(Wα) < ε.

(Definition of a set of measure zero is much more general, but for our purpose this analog in Rn is enough.)

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Annual Foundational School I, 2014, Almora

Tutorial Problems in Topology

1 Week 1, Prof. Archana S. Morye

1

Exercises

1. Prove that to every A ∈ (Rn)∗ corresponds a unique y ∈ Rn such that Ax = x · y. Provealso that ||A|| = |y|.

2. If x ∈ Rn, define φx ∈ (Rn)∗ by φx(y) = x · y. Define A : Rn → (Rn)∗ by Ax = φx.Show that A is one-one linear transformation and conclude that every φ ∈ (Rn)∗ is φx fora unique x ∈ Rn..

3. A linear transformation A ∈ L(Rn) is norm preserving if |Ax| = |x|, and inner productpreserving if Ax · Ay = x · y, for all x, y ∈ Rn. If x, y ∈ Rn are nonzero, the angle between

x and y, denoted ∠(x, y), is defined as arccos(

x·y|x||y|

). The linear transformation A is angle

preserving if A is one-one, and for all x, y 6= 0 we have ∠(Ax,Ay) = ∠(x, y).

(a) Prove that A is norm preserving if and only if A is inner product preserving.

(b) Prove that if A is norm preserving, then A is one-one, and A−1 is also norm preserving.

(c) Prove that if A is norm preserving, then A is angle preserving.

(d) If there is pairwise orthogonal basis {x1. . . . , xn} of Rn, i.e., xi · xj = 0, for all i 6= j,and real numbers λ1, . . . , λn such that Axi = λixi, for all 1 ≤ i ≤ n, prove that A isangle preserving if and only if |λi| are equal for all i, 1 ≤ i ≤ n.

(e) Prove that all angle preserving linear transformations A are of the form A = UV , whereU is angle preserving linear transformation of type (3d), and V is norm preservingtransformation.

(f) If 0 ≤ θ < π, let A : R2 → R2 be given by A

(xy

)=

(sin θ cos θ− cos θ sin θ

)(xy

). Show

that A is angle preserving, and if (x, y) 6= 0, then ∠((x, y), (Ax,Ay)) = π/2− θ.

4. If A ∈ L(Rn,Rm), then prove that A is continuous. Moreover if SA = {|Ax| |x ∈ Rn, |x| ≤1}, then show that SA = [0, ||A||].

5. Prove that ||A|| = min{λ ∈ R | |Ax| ≤ λ|x| for all x ∈ R}.

6. Let Y denote the space of all realvalued continuous functions on [0, 1]. Define a norm||.|| : Y → [0,∞) by ||f || = sup

x∈[0,1]|f(x)|, for f ∈ Y . Let X denote the subspace of Y

consisting of all functions f : [0, 1] → R that are continously differentiable on some openneighborhood Uf of [0, 1]. Give X the norm induced by Y . Define A : X → Y to be thelinear map Af = f ′, f ∈ X, where f ′ : [0, 1] → R is the derivative of f . Prove that A isnot continuous.

7. Prove that if f : Rm → Rn is differentiable at a ∈ Rm, then it is continuous at a.

8. If s : R2 → R is defined by s(x, y) = x+ y, then show that ds(a,b) = s.

9. If p : R2 → R is defined by p(x, y) = x · y, then show that dp(a,b)(x, y) = bx+ ay.

10. A function f : Rn ×Rm → Rp is bilinear if for x, x1, x2 ∈ Rn, y, y1, y2 ∈ Rm, and a ∈ Rwe have f(ax, y) = af(x, y) = f(x, ay), f(x1 + x2, y) = f(x1, y) + f(x2, y), f(x, y1 + y2) =f(x, y1) + y(x, y2).

2

Week 1, Prof. Archana S. Morye, continued

2

(a) Prove that if f is bilinear, then lim(h,k)→0

|f(h, k)||(h, k)| = 0.

(b) Prove that df(a,b)(x, y) = f(a, y) + f(x, b).

11. Let f : Rn → R be a function such that |f(x)| ≤ |x|2. Show that f is differentiable at 0.

12. Let f : R2 → R be defined by f(x, y) =√|xy|. Show that f is not differentiable at (0, 0).

13. Let g be a continuous realvalued function on S1 such that g(0, 1) = g(1, 0) = 0 and g(−x) =−g(x). Define f : R2 → R by

f(x) =

{|x| · g

(x|x|

), if x 6= (0, 0)

0, if x = (0, 0).

(a) If x ∈ R2 and h is defined by h(t) = f(tx), show that h is differentiable.

(b) Show that f is not differentiable at (0, 0) unless g = 0.

14. Let f : R2 → R be defined by

f(x, y) =

{x|y|√x2+y2

, if (x, y) 6= (0, 0)

0, if (x, y) = (0, 0).

Show that f is not differentiable at (0, 0).

15. Show that the function f(x, y) = |xy| is differentiable at (0, 0), but is not continouslydifferentiable in any neighborhood of (0, 0).

16. Use basic theorems (like Chain Rule, etc) to find differential of f(x, y) = sin(xy2).

17. Suppose f is a differentiable mapping of R into R3 such that |f(t)| = 1 for every t. Provethat f ′(t) · f(t) = 0. Interpret this result geometrically.

18. If f is a differentiable mapping of a connected open set E ⊂ Rn into Rm, and if dfx = 0 forevery x ∈ E, prove that f is constant in E.

19. If f(0, 0) = 0 and

f(x, y) =xy

x2 + y2if (x, y) 6= (0, 0),

prove that ∂f∂x1

and ∂f∂x2

exist at every point of R2, although f is not continuous at (0, 0).

20. Define f and g on R2 by: f(0, 0) = g(0, 0) = 0, f(x, y) = xy2

(x2+y4), g(x, y) = xy2

(x2+y6)if

(x, y) 6= (0, 0). Prove that f, g are not continuous at (0, 0), but partial derivatives of f andg exist at every point of R2.

21. Suppose that f is a realvalued function defined in an open set E ⊂ Rn, and that the partialderivatives ∂f

∂x1, . . . , ∂f

∂xnare bounded in E. Prove that f is continuous in E.

22. Suppose that f is a differentiable realvalued function in an open set E ⊂ Rn, and that fhas a local maximum at a point x ∈ E. Prove that dfx = 0.

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3

23. If f is a realvalued function defined in a convex open set E ⊂ Rn, such that ( ∂f∂x1

)(x) = 0for every x = (x1, . . . , xn) ∈ E, prove that f(x) depends only on x2, . . . , xn.

24. If f and g are differentiable realvalued functions in Rn, prove that ∇(fg) = f∇g + g∇f ,and that ∇(1/f) = −f−2∇f whenever f(x) 6= 0 for all x ∈ Rn.

25. Fix two real numbers a and b, 0 < a < b. Define a mapping f = (f1, f2, f3) of R2 into R3

by

f1(s, t) = (b+ a cos s) cos t

f2(s, t) = (b+ a cos s) sin t

f3(s, t) = a sin s

Describe the range K of f .

(a) Determine the zeros of ∇f1, i.e., {(s, t) | (∇f1)(s, t) = 0}.(b) Determine the set of all (s, t) ∈ R2 such that (∇f3)(s, t) = 0.

(c) Let S = {f(s, t) | (∇f1)(s, t) = 0} be a subset of K. Show that one of the points of Scorresponds to a local maximum of f1, one corresponds to a local minimum, and thatthe other two are neither (they are called saddle points). Also determine the set ofpoints of K which corresponds to a local maximum or minimum of f3

26. Let f : R → R2 be given by f(t) =

(cos tsin t

). Compute the Jacobian matrix of partial

derivatives for f and show that f(0) = f(2π) but f ′(t) 6= 0 for all t.

27. The mean value inequality. Suppose that U is an open set in Rn, and that f : U → Rm

is differentiable. Consider the straight line segment [x, y] = {(1−t)x+ty | 0 ≤ t ≤ 1} joiningx and y. If [x, y] ⊂ U and ||dfx|| ≤ K for all x ∈ [x, y], then |f(x)− f(y)| ≤ K|x− y|.

28. Let U = {(x, y) ∈ R2 | |(x, y)| > 1}\{(x, 0) |x ≤ 0}. Define θ : U → R as follows. For(x, y) ∈ U , θ(x, y) is the unique solution of

cos (θ(x, y)) =x

|(x, y)| , sin (θ(x, y)) =y

|(x, y)| , −π < θ(x, y) < π.

Show that θ is every where differentiable with ||dθ(x,y)|| < 1. Show, however, that if a =(−1, 1

10), b = (1, 1

10) then |θ(a) − θ(b)| > |a − b|. This exercise shows that we cannot

replace the line segment in the mean value theorem by other curves without changing theconclusion.

29. Show that a C1 map Rn → Rn takes sets if measure zero to sets of measure zero.

30. Prove that the image of a C1 map Rm → Rn has measure zero, if m < n.

31. Let f : GLn(R) → GLn(R) be given by f(A) = A−1. Prove that f is differentiable anddfA : L(Rn)→ L(Rn) is dfA(H) = −A−1HA−1.

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4

32. (a) Prove that the determinant function det : M(n,R) −→ R is differentiable, and

d(det)A(B) =n∑

i=1

det

A1...

Ai−1Bi

Ai+1...An

,

where for any matrix X, Xi denotes its i-th row.

(b) Identify the map d(det)I : M(n,R) −→ R, where I ∈ M(n,R) is the identity matrix.

(c) Show that Tr : M(n,R) −→ R is differentiable.

33. Matrix exponentials

(a) Define a sequence of functions fi : M(n,R) −→ M(n,R) by

f0(X) = I

fi(X) =X i

i!for i ≥ 1 .

Show that the series∑∞

i=0 fi converges uniformly on compact subsets of M(n,R).(Hint: ‖AB‖ ≤ ‖A‖‖B‖.)

(b) Thus, the definition

eX =∞∑

i=0

X i

i!

makes sense for all X ∈ M(n,R). Show that if A,B ∈ M(n,R) are such that AB =BA, then eA+B = eAeB. (Hint: Let Sk(X) =

∑kj=0 fj(X). Since matrix multiplication

M(n,R)×M(n,R) −→ M(n,R) is continuous, we have

eAeB = limk→∞

Sk(A)Sk(B) .

Estimate ‖S2k(A + B) − Sk(A)Sk(B)‖.) In particular, e−AeA = I, so eA ∈ GL(n,R)for every A ∈ M(n,R).

(c) Show that the map exp : M(n,R) −→ GL(n,R), X 7→ eX , is C∞, compute itsdifferential at 0, and show that it is a local diffeomorphism at 0. We call exp theexponential map on M(n,R).

34. Let f : R2 → R3 and g : R3 → R2 be given by the equations f(x1, x2) = (e2x1+x2 , 3x2 −cosx1, x

21 + x2 + 2), g(y1, y2, y3) = (3y1 + 2y2 + y23, y

21 − y3 + 1). If F = g ◦ f , find dF0. And

if G = f ◦ g, find dG0.

35. Let A ⊂ Rn be an open set and f : A→ Rn is a continuously differentiable one-one functionsuch that det(Jf(x)) 6= 0 for all x ∈ A. Show that f is open map, i.e., if B ⊂ A is open inA then f(B) is an open in Rn. Also show that f−1 : f(A)→ A is differentiable.

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5

36. Let f : R2 → R be a continously differentiable function. Show that f is not one-one.

37. (a) If f : R→ R satisfies f ′(a) 6= 0 for all a ∈ R, show that f is one one on all of R.

(b) Define f : R2 → R2 by f(x, y) = (ex cos y, ex sin y). Show that Jf(x, y) 6= 0 for all(x, y) but f is not one-one.

38. Use the function f : R→ R defined by

f(x) =

{x2

+ x2 sin 1x, x 6= 0

0, x = 0

to show that continuity of the derivative cannot be eliminated from the hypothesis of theinverse function theorem.

39. Show that the equations

ex + e2y + e3u + e4v = 4

ex + ey + eu + ev = 4

can be uniquely solved for u and v in terms of x and y around the point (0, 0, 0, 0).

40. LetS = {(x, y, z) ∈ R3 |xz + sin(xy) + cos(xz) = 1} .

Determine whether in the neighborhood of (0, 1, 1), S is the graph of a differentiable functionin any of the following forms

z = f(x, y) , x = g(y, z) , y = h(x, z) .

Archana S. Morye <[email protected]>

(End)

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6

AFS I 2014, CEMS, AlmoraTopology, Assignment 2

1 (Rolle’s theorem.) Let U be a non-empty bounded open subset of Rn, and letf : U → R be a continuous function which is constant on the frontier of U , anddifferentiable on U . Show that f has a critical point in U .

2 Prove that all norms on a finite-dimensional R-vector space V are equivalent, asfollows. Fix a basis {e1, . . . ,en} of V , and define a norm ‖·‖0 on V by

‖n

∑i=1

aiei‖0 = maxi|ai|.

Let ‖·‖ be an arbitrary norm on V .

(a) Show that there exists a constant C > 0 such that ‖v‖ ≤C‖v‖0 for all v ∈V .

(b) Prove that ‖·‖ is continuous on V with respect to ‖·‖0, and note that the set

S = {v ∈V |‖v‖0 = 1}

is compact with respect to the norm ‖·‖0.

3 Let n ∈ N. For each p ∈ [1,∞], define the p-norm ‖·‖p on Rn by

‖x‖p =

{(∑n

i=1‖xi‖p) 1p if 1≤ p < ∞,

max1≤i≤n‖xi‖ if p = ∞.

for all x = (x11, . . . ,xn) in Rn.

(a) Show that ‖x‖∞ = limp→∞‖x‖p for all x ∈ Rn.

(b) Check that each p-norm is indeed a norm on Rn.

4 Let U be an open subset of Rn, let x0 ∈ U , and let f : U → Rm be a map. Let‖·‖1 and ‖·‖2 be norms on Rn and Rm, respectively, that are equivalent to the usualnorms. Prove that f is differentiable at x0 if and only if there exists a linear mapA : Rn→ Rm such that limh→0

‖ f (x0+h)− f (x0)−Ah‖2‖h‖1

= 0.

5 Let U be an open subset of Rn, and let f : U → Rm be a C1 map. Show thatf is locally Lipschitz, that is, that for every point x0 in U , there exist an openneighbourhood U of x0, and a real number c> 0, such that ‖ f (x)− f (y)‖≤ c‖x−y‖for all x,y ∈U .

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7

2

6 Let f : Rn → R be the function f (x) = ‖x‖. Let x0 ∈ Rn \ {0}. Show that f isdifferentiable at x0, and that

d fx0(v) =1‖x0‖〈x0,v〉

for all v ∈ Rn, where 〈·, ·〉 denotes the inner product on Rn.

7 Let U be an open subset of Rn, and let f : U → Rn be a C1 map. Suppose that themap d fx : Rn→ Rn is an isomorphism for every point x in U .

(a) Prove using the inverse function theorem that f is an open map.

(b) Show that if f is injective, then it is a diffeomorphism from U onto the opensubset f (U) of Rn.

8 Prove that a C1 map from R2 to R is not injective.

9 Let Mn(R) be the space of all n×n real matrices.

(a) Prove that the determinant function det : Mn(R)→ R is differentiable, and

d(det)A(B) =n

∑i=1

det

A1...

Ai−1Bi

Ai+1...

An

,

where for any matrix X , Xi denotes its i-th row.

(b) Identify the map d(det)I : Mn(R)→ R, where I ∈Mn(R) is the identity ma-trix.

10 Matrix exponentials.

(a) Define a sequence of functions fi : Mn(R)→Mn(R) by

f0(X) = I

fi(X) =X i

i!for i≥ 1.

Show that the series ∑∞i=0 fi converges uniformly on compact subsets of

Mn(R).

(Hint: ‖AB‖ ≤ ‖A‖‖B‖.)

Topology, Assignment 2


8

3

(b) By (a), the definition

eX =∞

∑i=0

X i

i!

makes sense for all X ∈ Mn(R). Show that if A,B ∈ Mn(R) are such thatAB = BA, then eA+B = eAeB. In particular, e−AeA = I, so eA ∈ GLn(R) forevery A ∈Mn(R).

(Hint: Let Sk(X) = ∑kj=0 f j(X). Since matrix multiplication Mn(R) ×

Mn(R)→Mn(R) is continuous, we have

eAeB = limk→∞

Sk(A)Sk(B) .

Estimate ‖S2k(A+B)−Sk(A)Sk(B)‖.)

(c) Show that the map exp : Mn(R) → GLn(R), X 7→ eX , is C∞, compute itsdifferential at 0, and show that it is a local diffeomorphism at 0. We call expthe exponential map on Mn(R).

11 Show that the equations

ex + e2y + e3u + e4v = 4ex + ey + eu + ev = 4

can be uniquely solved for u and v in terms of x and y around the point (0,0,0,0).

12 LetS = {(x,y,z) ∈ R3 |xz+ sin(xy)+ cos(xz) = 1}.

Determine whether in the neighbourhood of (0,1,1), S is the graph of a differen-tiable function in any of the following forms

z = f (x,y), x = g(y,z), y = h(x,z).

13 Let U be a convex open neighbourhood of 0 in Rn, and let f : U → R be a C∞

function such that f (0) = 0. Show that there exist C∞ functions g1, . . . ,gn from Uto R such that

(a) f (x) = ∑ni=1 xigi(x) for all x = (x1, . . . ,xn) ∈U ; and

(b) gi(0) =∂ f∂xi

(0) for all i = 1, . . . ,n.

Hint: Note that

f (x) =∫ 1

0

d fdt

(tx)dt .

Topology, Assignment 2


9

Annual Foundation School - 1CEMS, Almora

Topology : Assignment - 3

(1) Let (X, d) be a metric space. Define

δ(x, y) =d(x, y)

1 + d(x, y)

for all x, y ∈ X. Show that δ is a metric on X.

(2) Show that the countable collection

B = {(a, b) | a < b, a and b rational}is a basis that generates the standard topology on R.

(3) Show that the collection

C = {[a, b) | a < b, a and b rational}is a basis that generates a topology different from the lower limit topology on R.

(4) Show that if A is closed in X and B is closed in Y , then A×B is closed in X × Y .

(5) A family (Ai)i∈I of subsets of a topological space X is said to be locally finite if for each

x ∈ X there is a neighbourhood V of x such that V ∩ Ai = ∅ for all but finite number

of indices i ∈ I. Show that the union of a locally finite family of closed subsets of a

topological space X is closed in X.

(6) Let A,B, and Aα denote subsets of a space X. Prove the following:

(a) If A ⊂ B, then A ⊂ B.

(b) A ∪B = A ∪B.

(c)⋃Aα ⊃

⋃Aα; give an example where equality fails.

(7) If L is a straight line in the plane, describe the topology on L inherited as a subspace of

R` × R and as a subspace of R` × R`.

(8) Show the T1 axiom is equivalent to the condition that for each pair of points of X, each

has a neighbourhood not containing the other.

(9) Let f : A → B and g : C → D be continuous functions. Define a map f × g : A × C →B ×D by the equation

(f × g)(a× c) = f(a)× g(c).

Show that f × g is continuous.

(10) Let X be a non-empty set. Let a ∈ X be a fixed point and set

T = {∅} ∪ {U ⊂ X | a ∈ U}(a) Check that T is a topology on X.

(b) Is T Hausdorff?

(c) Find limit points of {a}.(d) Deduced that non empty open sets are dense in X.

1

2 Week 2, Prof. Sanjay H. Amrutiya

10

2

(e) Let A be any set containing a. Prove that the set of limit points of A is X − {a}.(f) Show that every proper subset of X − {a} is no where dense, that is,

◦A = ∅.

(g) What is subspace topology on X − {a}? Is it Hausdorff?

(11) Here is a proof of Euclid’s theorem on the infinitude of primes, due to H. Furstenberg

(1955).

(a) Prove that the sets a+ nZ ( a, n ∈ Z, n ≥ 1 ) form a basis for a topology T in Z.

(b) Prove that each of the above sets is closed with respect to T .

(c) Prove that Z\{−1, 1} is the union of pZ as p runs over all prime numbers.

(d) Conclude that there are infinitely many primes.

(12) Show that if Y is compact space, then the projection π1 : X × Y → X is a closed map.

(13) Let f : X → Y be a map, let Y be a compact Hausdorff space. Then f is continuous if

and only if the graph of f ,

Gf = {(x, f(x)) : x ∈ X}

is closed in X × Y .

(14) Let fi : Xi → Yi (i ∈ I) be a family of open maps, and suppose that fi is surjective for

all but finitely many indices i. Show that the product map

f :∏

i∈IXi →

∏

i∈IYi

(xi) 7→ (fi(xi)),

is an open map.

(15) Let Y be an ordered set in the ordered topology. Let f, g : X → Y be continuous.

(a) Show that the set {x ∈ X | f(x) ≤ g(x)} is closed in X.

(b) Let h : X → Y be the function

h(x) = min{f(x), g(x)}.

Show that h is continuous.

(16) Show that every metrizable space with countable dense subset has a countable basis.

Conclude that Rl is not metrizable.

(17) Show that every metrizable Lindelof space has a countable basis.

(18) Show that a closed set of a normal space is itself normal space.

(19) Give a direct proof of Urysohn lemma for a metric space (X, d).

(20) Let X be a compact Hausdorff space. Then show that C(X), the set of all real valued

continuous functions on X, is finite dimensional if and only if X is finite.

(21) Consider the space X = R with co-countable topology.

(a) Determine the connected and dis-connected subspaces of X.

(b) Determine the compact subspaces of X.

Week 2, Prof. Sanjay H. Amrutiya, continued

11

3

(22) Consider the space X = R with co-countable topology. Let f : R → X be a continuous

function, where R is with usual topology on it. Show that f is constant. What can you

say about a continuous function f : X → R ?

(23) Show that if X is Lindelof and Y is compact, then X × Y Lindelof.

(24) Let p : X → Y be a closed map.

(a) If p−1(y) ⊂ U , where U is an open subset of X, then p−1(W ) ⊂ U for some neigh-

bourhood W of y in Y .

(b) If p−1(A) ⊂ U for some subspace A of Y and some open subspace U of X, then

p−1(W ) ⊂ U for some open neighbourhood W of B in Y .

(25) Let p : X → Y be a closed continuous surjective map such that p−1(y) is compact, for

each y ∈ Y . Show that p−1(C) is compact for any compact subset C of Y . In particular,

if Y is compact, then X is compact.

(26) Let p : X → Y be a closed continuous surjective map such that p−1(y) is compact, for

each y ∈ Y .

(a) Show that if X is Hausdorff, then so is Y .

(b) Show that if X is regular, then so is Y . Show that if X is locally compact, then so

is Y . Show that if X is second-countable, then so is Y .

(27) Show that every locally compact Hausdorff space is completely regular.

(28) Let X be a compact Hausdorff space. Show that X is metrizable if and only if X has a

countable basis.

(29) Show that every locally compact Hausdorff space is completely regular.

(30) Show that there is no continuous function f : R → R such that f(Q) ⊂ R\Q and

f(R\Q) ⊂ Q.

(31) Let N be endowed with the co-finite topology. Show that N is not path connected.

(32) Show that GLn(R) is not connected. What can you say about GLn(C)?


12

AFS I 2014, CEMS, AlmoraTopology, Polar decomposition

1 The exponential map exp : Mn(C)→ GLn(C) is defined by exp(A) = ∑∞i=0

1i!A

i.The real version of this map was discussed in Assignment 2. The proof of theconvergence of the above series in the complex case is an exact analogue of theproof in the real case.

(a) Prove that if A is a diagonal matrix in Mn(C) with diagonal entries α1, . . . ,αn,then exp(A) is a diagonal matrix with diagonal entries eα1 , . . . ,eαn .

(b) Prove that exp(TAT−1) = T exp(A)T−1 for all A ∈Mn(C) and T ∈ GLn(C).

2 Recall the following facts from linear algebra.

(a) All the eigenvalues of a Hermitian matrix are real.

(b) A Hermitian matrix is positive definite if and only if all its eigenvalues arepositive. (The positivity makes sense because, by (a), all the eigenvalues arereal.)

(c) For every n× n Hermitian matrix A, there exists an n× n unitary matrix Tsuch that the matrix TAT−1 is diagonal.

Prove that for every n×n positive definite Hermitian matrix P, there exists a uniquen×n Hermitian matrix H such that exp(H) = P. We call H the logarithm of P, anddenote it by log(P).

3 Prove that for every n×n positive definite Hermitian matrix P, and for every integerr ∈ N, there exists a unique positive definite Hermitian matrix H such that Hr = P.We call H the r-th root of P, and denote it by r

√P.

4 Let PH(n) be the subset of Mn(C) consisting of positive definite Hermitian ma-trices, and U(n) the subgroup of GLn(C) consisting of unitary matrices. LetA ∈ GLn(C).

(a) Prove that the matrix AA∗ is positive definite and Hermitian.

(b) Let P =√

AA∗, and U = P−1A. (The definition of A makes sense by (3).)Prove that U is unitary.

(c) Prove that (P,U) is the unique element of PH(n)×U(n) such that A = PU .The pair (P,U) is called the polar decomposition of A.

5 Let (Ai)i∈N be a bounded sequence in Mn(C), and suppose that all the convergentsubsequences of (Ai) have the same limit A. Prove that (Ai) converges to A.

1


13

2

6 Let H(n) denote the R-subspace of Mn(C) consisting of Hermitian matrices. Provethat the map exp : Mn(C)→GLn(C) restricts to a homeomorphism from H(n) ontoPH(n).

7 Let r ∈ N. Prove that the map A 7→ r√

A from PH(n) to itself is continuous.

8 Prove that the map (P,U) 7→ PU from PH(n)×U(n) to GLn(C) is a homeomor-phism.

9 Prove that U(n) is a strong deformation retract of GLn(C).

Topology, Polar decomposition


14

TopologyAssignment - 5, AFS-I 2014

CEMS, Almora

1. Show that if X is a second countable space then every basis of X contains a countable basis.

2. Let f : X → Y be a continuous open map. Show that if X satisfies the first or secondcountability axiom, then f(X) satisfies the same condition.

3. Show that if X has a countable dense subset, every collection of disjoint open sets in X iscountable.

4. Show that the cone C(X) over any topological space X is contractible.

5. Let F : X → Y be a continuous map. Show that f is nulll homotopic if and only if thereexists a continuous map f : C(X)→ Y which extends f .

6. Define an equivalence relation on plane R2 as follows :

(x0, y0) ∼ (x1, y1) if x20 + y20 = x21 + y21.

Identify the quotient space with some familiar space.

7. Define an equivalence relation on Euclidean space R as follows:

r ∼ s if (r − s) ∈ Q.

Show that quotient space is indiscrete topological space. Conclude that quotient space ofHausdorff topological space need not be Hausdorff.

8. Let π1 : R2 → R be projection on the first coordinate. Let A be the subspace of R2

consisting of all points (x, y) for which either x ≥ 0 or y = 0 (or both); let q : A → R beobtained by restricting π1. Show that q is quotient map that is neither open nor closed.

9. Let X have a countable basis; let A be an uncountable subset of X. Show that uncountablymany points of A are limit points of A.

10. Show that every compact metrizable space X has a countable basis.

1

3 Week 3, Prof. Umesh V. Dubey

15

TopologyAssignment - 6, AFS-I 2014

CEMS, Almora

1. Let R be the equivalence relation on R obtained by identifying all the points of Z. Showthat R is closed and R/Z is Hausdorff but not locally compact.

2. Let R be the following equivalence relation on X := [−1, 1] : if x 6= 1 or −1 then x ∼ −x.Show that R is open but not Hausdorff. Also, show that the quotient space X/R can beobtained by pasting together two normal spaces along open subsets.

3. Show that mapping cone of f : S1 → S1; z 7→ z2 is real projective plane. Also, show thatmapping cylinder of f is Mobius band.

4. Prove that Dn/Sn−1 is homeomorphic to Sn where Dn is unit ball in n-dimentional Euclideanspace Rn.

5. Show that Mobius band and cylinder are of same homotopy type but not homeomorphic.

6. Identify the coset space (R,+)/Z with some familiar topological group.

7. Prove that every nontrivial discrete subgroup of R (resp. S1) is infinite cyclic (resp. finitecyclic).

8. Prove that O(n) is homeomorphic to SO(n)× Z2. Are these two isomorphic as topologicalgroups.

9. Let G be a topological group.

(a) Let A and B be subspaces of G. If A is closed and B is compact, show A ·B is closed.

(b) Let H be subgroup of G; let p : G → G/H be the quotient map. If H is compact,show that p is a closed map.

(c) Let H be a compact subgroup of G. Show that if G/H is compact, then G is compact.

10. Let G be a compact topological group; let X be a topological space; let α be an action ofG on X. If X is Hausdorff, or regular, or normal, or locally compact, or second countable,so is the orbit space X/G.

1

Week 3, Prof. Umesh V. Dubey, continued

16



Topology:Assignment VII

December 22, 2014

1. Show that for a subset A of a topological space X, the following conditions are equivalent:

(a) There exist an open subset U , and a closed subset F of X such that A = U ∩ F .

(b) A is open in its closure A.

(c) For every point a ∈ A, there exists an open neighbourhood Ua of a in X such thatUa ∩ A is closed in Ua.

If A satisfies any one of the above conditions then it is called locally closed.

2. Prove that every locally closed subspace of a locally compact space is locally compact.

3. If X is a Hausdorff topological space and Y is a dense locally compact subset of X, thenshow that Y is open.

4. Show that, the following conditions on a topological space X are equivalent:

(a) For every point x ∈ X, the set of closed neighbourhoods of X is a fundamental systemof neighbourhoods of x.

(b) For every closed subset F of X, and for every point x ∈ X \ F , there exist openneighbour- hoods UF and Ux of F and x, respectively, such that UF ∩ Ux = ∅.

A topological space space X is said to be regular if it is Hausdorff, and satisfies one of theabove conditions.

5. Let X be a locally compact space. Then prove the following statemets:

(a) The space X is regular.

(b) Every point in X has a fundamental system of compact neighbourhoods.

(c) The space X is compactly generated, that is, a subset F of X is closed if and only ifF ∩K is closed for every compact subspace K of X.

6. Prove from the the definition that Rn is paracompact.

7. Show that discrete and indiscrete topological spaces are alway paracompact.

8. If X is a paracompact topological space, and K a compact topological space, then provethat X ×K is paracompact.

9. Show that the product of paracompact topological spaces need not be paracompact.

10. Let X be a paracompact topological space, and A ⊂ X closed. Then show that X/A isparacompact.

4 Week 4, Prof. N. Raghavendra

17



Topology:Assignment VII

December 25, 2014

1. Bump functions

(a) Show that the function f : R −→ R

f(t) =

{exp (−1/t) if t > 0 ,

0 if t ≤ 0

is C∞. Does it have a Taylor series expansion around the point t0 = 0?

(b) Let a ∈ Rn and let 0 < δ < ε. For any r > 0, denote by B(a, r) the open ball in Rn

with centre a and radius r. Show that there exists a C∞ function f : Rn −→ R suchthat 0 ≤ f ≤ 1 on Rn, f = 1 on the closed ball B(a, δ) and f = 0 outside B(a, ε).

(c) If K ⊂ Rn is compact and U is an open neighbourhood of K in Rn, then there existsa C∞ function f : Rn −→ R such that 0 ≤ f ≤ 1, f = 1 on K and f = 0 outside U .

2. Let U and V be open subsets of Rn and Rm respectively. Show that C∞ homotopy is anequivalence relation on the set of C∞ maps from U to V .

3. Let σ : [0, 1] −→ U ⊂ Rn be continuous path. Show that there exists a C∞ path τ : R −→ Usuch that σ is homotopic to τ relative to the set {0, 1}.

4. Let σ : [0, 1] −→ S2 be a continuous path. Then prove the there exists a C∞ path τ : R −→S2 such that σ is homotopic to τ relative to the set {0, 1}. Recall that a map f : R −→ S2

is C∞ if the composition i ◦ f : R −→ R3 is C∞, where i denote the canonical inclusion ofS2 into R3.

5. Prove S2 is simply connected using the Sard theorem.

6. Prove the Sard theorem for one variable case.

Week 4, Prof. N. Raghavendra, continued

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