Let N be endowed with the discrete topology and Y = (0} U<,nEN-(0.1)> be a subspace ofR. The topology on Y is the induced topology by the Euclidean topology on R. We define the functionS:N-(0}- Y by f(1) = 0 and f(n)=Vn > 1.1. is a one to one function.a Trueb. False2. f is onto.a Trueb. False3. f is continuous.a. Trueb. False4. (0} is open in Y.a. Trueb. False5. is continuous.a. Trueb False6. f is a homeomorphism.a. Trueb. False

Answered: Image /qna-images/answer/67cd7b9c-50d0-4554-a0f6-2e7be5fd24e7.jpg

Let N be endowed with the discrete topology and Y = (0} u, nEN- (0, 1}> be a subspace of R. The topology on Y is the induced topology by the Euclidean topology on R. We define the function f:N-(0} Y by f(1) 0 and f(n) = , %3D Vn > 1. 1. is a one to one function. a True b. False 2. f is onto. a True b. False 3. f is continuous. a. True b. False 4. (0} is open in Y. a. True b. False 5. is continuous. a. True b. False 6. f is a homeomorphism. a. True b. False

Let N be endowed with the discrete topology and Y = (0} u, nEN- (0, 1}> be a subspace of R. The topology on Y is the induced topology by the Euclidean topology on R. We define the function f:N-(0} Y by f(1) 0 and f(n) = , %3D Vn > 1. 1. is a one to one function. a True b. False 2. f is onto. a True b. False 3. f is continuous. a. True b. False 4. (0} is open in Y. a. True b. False 5. is continuous. a. True b. False 6. f is a homeomorphism. a. True b. False

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

11-please fast
Suppose that W is a subspace of normed space V then it is not necessary that W is also normed space. 13- True False
20. Let f be a continuous linear functional defined on the normed linear space X. Show that M = {xe X\f(x) = 0) is a closed subspace of X.
7. (**) Let V be a vector space over C of dimension n. Let X be the vector space of linear transformations from V to V; denote by 0 the zero map on V, which is the "zero vector" in the vector space X. Also, X contains the identity map I: V→ V. Let L: V→V be a linear transformation (so that I can be viewed as a vector in X). Further, L²= LoL EX and L³ = Lo Lo € X, etc. We define Lº := I. If q(t) is a polynomial with coefficients in C, then q(L): V → V is defined to be the following linear transformation: q(L) = apI+a₁L + a₂L²+...+ am L™ where q(t) = ao+a₁t+...+amt™ that is q(L) is just a linear combination of the powers of L, so q(L) is a "vector" in X. Assume that L 0. Prove the following: a) Using the fact that X is a vector space of dimension n² over C, show that the set of powers of L must be linearly dependent. b) Using (a), show there exists a polynomial q(t) such that q(L) = 0, the zero map. c) Show that there exists a least integer k such that there is a monic polynomial q(t) of…
Find a basis for the space of 2 by 3 matrices whose nullspace contains (2, 1, 1 ).
Please don't use generated AI
5. (*) Let V be a vector space over F and let U be a subspace of V. Define a relation ~ on V by v~wv-w€U (a) Prove that is an equivalence relation on V. Write [v] for the equivalence class of containing v € V. (b) Show that [v] = {v+u: u € U}. For u, w € V we define [v] + [w] = [v+w]~ and for λ E F we define X[v] = [v]. (c) With this definition of addition and scalar multiplication, prove that {[v]~: v € V}, the set of equivalence classes of ~, is a vector space over F. (d) Take V = R³, U = -{(+₁)=zex}. : ER and define as above. Find a set of representatives for the equivalence classes of~. (e) With V and U as in part (d), (c) tells us the set of equivalence classes of ~ is a vector space over R. What is the dimension of this vector space?
4. Let V and W be finite dimensional vector spaces. Let T : V –→ W be a linear transformation of V into W. (a) Let H be a subspace of V. Let T(H) be the set of all images of the vectors in H under the transformation T. Show that T(H) is a subspace of W and dim T(H) < dim H (b) Now suppose that T is one-to-one. Show that dim T(H)= dim H.
Determine if the given set is a subspace of the appropropriate R^n and justify your conclusion.
Suppose H is a k-dimenisonal subspace of R". Then dim(H+) = n − k. True O False
(see attached)
4) Please correctly and detailed justification are required