Q3: (a) Give an open covering of the following sets which has no finite subcovring(D The open interval (0. 1) in (R.H).() R in (R,).(b) Determine which of the following sets are compact in (R.1) and give reason(i) z.(ii) (0, 1)U[3,6).

Answered: Image /qna-images/answer/47ee09de-54e2-4c17-bee4-2b3eaac1c00c.jpg

Answered: Q3: (a) Give an open covering of the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Q1) Choose the correct answer 1- Let X be a non-empty set and r = (-0,a)| a € R) URU(4) then r is called: (11) Indiscrete (1) Discrete (I) Left ray (IV) Right ray 2- Let X be a topological space and AS X, then the largest open set contained in A is: (1) Ext (A). (1) A". (IV) Ā (1) A'. 3- Two subsets A and B of the space X are called pair of disconnection of X if (1) AnB = 6, AUB = x () AnB + (m) AUB = X 4- Let X = (a, b, c), and 1= (X, , [a), (b,e}}, then the subset {c) is : (1) open and closed. (IV) All. (1) open. (I) closed. (IV) not open not closed Q2) True or false (1) B = {[a), (b), (c} is a base of the discrete topology on X = (a, b,c). (2) r= {x,. (x,y). (x,z). [y.z)} is not a topology on X = {x, y,z}. (3) For X = (a, b, c), the excluding point topology is E, = {X, g. (a), (c). (a, c), (a. b}). (4) For a discrete topology on X and o + ACX, the cdosure of A is A. (5) For an indiscrete topology on X = {a, b, c) and A = {a, b), the limit points of A is (a, b). (6) For the usual…
(iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A)
Q.3: let X= R, A-[0,] and TA) = {0,A,x] (3-Get topelog4) Q Frnd all the closed Sets inB, TCA)) () Find the closure, interior and baundary of the sets: S- [-,] M: [2, 3] %3D
3) Let X= {a, b, c, p} be a topological space with the particular point topology (where p is the particular point). Let A= {b,p}. Compute the following: (i) the open sets (iii) the closure of A limit points of A а, d: (ii) the closed sets (iv) the interior of A (v) ô(A) (vi) the
Show that A and B are compact sets A:={(x, y) E R^2: x^2 + y^5 = 2} B:= (x, y) E R2 : a² + 8xy + y < 1
Let R be the relation on the set A (1,2,3) defined by R= ((1,1),(1,3),(3. 1). (3,3)), then ROR--R OA R-(1().(4,4).19,9) R-R Ris antidymmetric Wind
1.54. (!) Let S = {(x, y) e R²: y ≤ x and x + 3y ≥ 8 and x ≤ 8). a) Graph the set S. b) Find the minimum value of x + y such that (x, y) e S. (Hint: On the graph from part (a), sketch the level sets of the function f defined by f(x, y) = x + y.)
Let x = (x1, x2, x3) ER³ and suppose you are given a subset UCR³ defined by U = {ER³ : x₂ < 5} Which of the following statements are necessarily true? (i) The complement of U contains a boundary point. (ii) The complement of U contains no interior points. (iii) U is not an open set. (iv) U is not a closed set. (A) (iii) only (B) (i) and (iv) (C) (ii) only (D) (i), (iii), and (iv) (E) (ii) and (iii) (F) (iv) only (G) (i) only (H) (iii) and (iv)
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Q3: (a) Give an open covering of the following sets which has no finite subcovring (D The open interval (0, 1) in (R.H). () R in (R,). (b) Determine which of the following sets are compact in (R.1) and give reason (i) z. (ii) (0, 1)U(3,6).