(a) Let X be a space, f: S' + Xa continuous map. Show that f is nullhomotopic (i.e. homotopic to a constant map) if and only if thereis a continuous map g: D2X with gIS' f. (Hint: If e is a con-stant map and F: cf then define g(rx) F(x,r) for x ES',rEIand use Exercise 8.14(f).)(b) Let x, y € X. Denote by P(x,y) the set of equivalence classes of

Answered: Image /qna-images/answer/56e16477-b816-4b0a-806e-a2361b96e352.jpg

Answered: (a) Let X be a space, f: S' +Xa…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 27E: 27. Let , where and are nonempty. Prove that has the property that for every subset of if and...

See similar textbooks

Similar questions

27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .
Bolzano-Weirstrass theorem for ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. Conclude that, with this metric, a subset of ℝ3 is sequentially compact if and only if it is closed and bounded.
let (X,T) be a topological space. Then a function f is continuous at x0 element of X if and only if f is both lower semi continuous and upper semi continuous at x0 element of X.
Let (X, T) and (Y, T1) be two topological spaces and let f be a continuous mapping of X into Y. If (Y, T1) is a T1 space, then (X, T) is a T1 space?
Consider the Banach space C[0,1] of continuous functions on the interval [0,1] equipped with the sup-norm. Let T: C[0,1] -> C[0,1] be a bounded linear operator such that T(f) is continuously differentiable for every f in C[0,1]. Prove or disprove the following statement: "If T is injective, then T^{-1} is also bounded."
5. Choose whether each of the following statements are true or false.(b) Let (Mi,di) metric spaces for i = 1,2 and f : M1 → M2 be continuous. Thenf(U) is open in M2 for all open subset U in M1. (c) Every homeomorphism is uniformly continuous. (d) Every contraction mapping has a fixed point.(e) An identity function on any metric space is a contraction.(f) A surjective isometry is a homeomorphism. (g) If f,g : R→R are uniformly continuous function on R then the produc
Let T be a linear operator on a finite-dimensional inner product space V. (a) If T is an orthogonal projection, prove that ||T(x)||≤||x|| for all x ∈V. Give an example of a projection for which this inequality does not hold. What can be concluded about a projection for which the inequality is actually an equality for all x∈V? (b) Suppose that T is a projection such that ||T(x)||≤||x||for x ∈V.Prove that T is an orthogonal projection.
Let X and Y be topological spaces and endow X×Y and Y×X with the respective product topologies. Show that the map f:X×Y→Y×X defined by f(x,y)=(y,x) is a homeomorphism.
This is a real analysis question. Let (X,d) be a complete metric space with X not ∅. Suppose the function f : X → X has the property that there exists a constant C ∈ (0, 1) such that d(f(x), f(y)) ≤ Cd(x, y) for all x,y ∈ X. The goal of this problem is to prove that there exists a unique x^∗ ∈ X satisfying f(x^∗) = x^∗. (a) Prove that f : X → X is continuous. (b) Prove that if p, q ∈X satisfy f(p ) = p and f(q) = q, then p = q. (This establishes uniqueness.)
This is a real analysis question. Let (X,d) be a complete metric space with X not ∅. Suppose the function f : X → X has the property that there exists a constant C ∈ (0, 1) such that d(f(x), f(y)) ≤ Cd(x, y) for all x,y ∈ X. The goal of this problem is to prove that there exists a unique x^∗ ∈ X satisfying f(x^∗) = x^∗. (a) Prove that f : X → X is continuous. (b) Prove that if p, q ∈X satisfy f(p ) = p and f(q) = q, then p = q. (This establishes uniqueness.) (c) Let x0 ∈ X be arbitrary. If the sequence {xn, n ∈ N} is defined by setting xn = f(xn−1) for n ∈ N, prove that {xn, n ∈ N} is Cauchy. (d) Since the metric space (X, d) is assumed to be complete, define the limit of the sequence {xn, n ∈ N} from (c) to be x^∗. Prove that f(x^∗) = x^∗. (This establishes existence.)
Let (X1, d1) and (X2, d2) be separable metric spaces. Prove that product X1 × X2 with metric d((x1, x2), (y1, y2)) = max{d1(x1, y1), d2(x2, y2)} is also separable space.
Let D = {(x, y) : x2 + y2 ≤ 1} be the unit disk in the plane. Place an equivalence relation ∼ on D by (x1, y1) ∼ (x2, y2) IFF x12 + y12 = x22 + y22. a) Find a continuous function. f:D → R (where R is the real number line) such that f((x1, y1)) = f((x2, y2)) IFF (x1, y1) ∼ (x2, y2) b) What space is D/~ homeomorphic to and why?

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

(a) Let X be a space, f: S' +Xa continuous map. Show that f is null homotopic (i.e. homotopic to a constant map) if and only if there is a continuous map g: D2X with gIS' = f. (Hint: If e is a con- stant map and F: c=f then define g(rx) = F(x,r) for x ES',rEI and use Exercise 8.14(f).)