Do the following task. a) State formally the Heine-Borel theorem in R^k b) Apply Heine-Borel theorem to show that if A is bounded, B is closed such that B ⊆ A ⊆ R^k, then B is compact. c) Give an example of set A and B in R such that A is bounded, B is closed, B ⊆ A but A is not compact.

Do the following task. a) State formally the Heine-Borel theorem in R^k b) Apply Heine-Borel theorem to show that if A is bounded, B is closed such that B ⊆ A ⊆ R^k, then B is compact. c) Give an example of set A and B in R such that A is bounded, B is closed, B ⊆ A but A is not compact.

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

Related questions

Q: Ametric space (x,d) is complete if and only ifx has the cantor intersection property

Q: 1/ Let (X₁T) and (Y₁J.) be top spaces and fix-x be a surjective and continuousmap, then y is Compact…

Q: 2. (a) Use axiom (2) in the definition of a topology to show that if {U1,U2, . ,Un} is a finite…

A: It is given in part (a) that (X,t) be any topological space, and the interval (x-A,x+A) is only open…

Q: det (NT) is sPace Toflogecu! and T=0U{un}such that 7Show that (N, Cl is compact slase

Q: Let's say that a collection of open balls (Ui)ie1 = (Bx, (ri))iel in Euclidean space Rd is = UB₂,…

Q: 4: state and prove L is a normed space ?

A: To prove that the space of bounded sequences, denoted as L^∞, is a normed space,we need to show that…

Q: Question Show closed bale Y in Mebic space (X,d), Alsa ఎవక is set that ? then shaw is. complete d)is…

A: Given: Y is closed ball in metric space (X, d). To show: 1. Y is closed set in (X, d). 2. If (X, d)…

Q: Let (R,d) be diserete metric space then R is not compact . True or false.??

A: Let (R,d) be diserete metric space then R is not compact . True or false.??

Q: For each of the following relations ~ on sets X, determine whether it is reflexive, symmetric or…

Q: 25. For any set S CR, let S denote the union of all the open sets contained in S. (a) Prove that S…

A: The given question is solved below.

Q: Other Math Yes a = [5 1], b = [1 4], c = [15 3], d = [8 13] Classify each of the following…

Q: Show that finite dimensional inner product space is Hilbert space

A: We are to show that finite dimensional inner product space is Hilbert space.

Q: be nonempty Subsets of Space X. Then Prove Let A and BJA inner product an TT 10) A CAL TT %3D (C) A…

Q: (BOunded st) show that a subset M in a nor med space Xis bounded if and only If there is a positive…

Q: 1) Let S be the equivalence relation on R defined by xSy iff x-y Q. Then a) [√3]-(-√3+r,req) b)…

Q: n topulogical manikuld - show that Mis locally compact 6

A: This is a problem of differential geometry.

Q: A subset (a,b) ≤R is T2-space in (R,Tu) * 1 true 2 False

A: Given: A subset (a, b)⊆ℝ is T2-space in ℝ, Tu. To determine: Whether given statement is true or…

Q: With the help of exomples Show thaat on finite dimensional are equivalent. any two norms Linear…

A: Given: Any two norms on finite dimensional linear space are equivalent.

Q: prove that i } X is a Coun table set and P: X>Y is asurjecbive then Y is Countable

A: Note: X denotes cardinality of X, that is number of elements in the set X. If X is countable means…

Q: 12 Let (RTR) is topological Space Such that TR-243 U2GSR=R-G-G is finste} Find buz), Z², Z. z' then…

A: Here given ℝ, Tℝ is a topological space such that Tℝ=ϕ∪G⊂ℝ: ℝ-G=Gc is finite This topology is known…

Q: Show the the Sar induce

A: According to the given information, it is required to show that equivalent norm on a vector space X…

Q: ctors - Z. S. Madhi

Q: True or False: Consider the subsets (0, 1) and (2, 3) of R, the set of all real numbers with the…

A: The given statement is Consider the subsets (0, 1) and (2, 3) of R, the set of all realnumbers with…

Q: Show that close ball Y in a metfic space (X,d) is a closed set also show that if (X,d) is complete…

A: Show that closed ball Y in a metric space (X,d) is a closed set also show that if (X,d) is complete…

Q: Prove that a finite projective plane of order n has n2 + n + 1 points and n2 + n + 1 lines.

A: Let p be an ambitrany point, each point in it determinis a line together with P which Obviously…

Q: on 'N" en Prove Delween Aborms is lenre equala bub

Q: 2. Determine which spaces under the given operatons. a)the set of all real- valued functions f…

Q: Exercise 10: Show that if A and B are bounded subsets • bounded set. Show that then AUB is a of IR,…

Q: 1/ Let (X₁T) and (Y₁J.) be top spaces and fix-x be a surjective and continuousmap, then y is Compact…

Q: Show that ([0,1] spack по matric matriz, is complete uniform norm

Q: / Let (X₁T) and (Y₁J) be top. Spaces and fix-x be a surjective and continuous map, then y is Compact…

Q: determine an b) Every Space CD Every non True or False, Justify why, not totally bounded Open Set S…

Q: Let (M, d) be a metric space and K ⊂ M. The following statements are equivalent: (i) K is relatively…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Exercise 1. Give examples of subsets A, B in a topological space X (for example X = R or X = R²)…

A: Disconnected sets

Q: Consider relation R={(x,y)|x divides y,xEA,y€A} on the set A={2,3,4,5,10,12,25,30}. (a) Check…

A: We are given a relation : R={(x,y)|x divides y,x∈A,y∈A} on a set A={2,3,4,5,10,12,25,30}

Q: Exercise 1. Give examples of subsets A, B in a topological space X (for example X = R or X = R²)…

Q: Prove from fundamental definitions and principles, that (-00, 0] and (0, 4] are numerically…

Q: Prove or disprove the following statement: A) Every T'3-space is T4-space?( explain your answer)

A: The statement "Every T3-space is a T4-space" is true. To understand why, let's first define the…

Q: consider a relation R={(a,b)|0<=a-b<=1} on a set A={0,1,2,3,4} a) Describe R using masters method…

Q: Determine the dimensions of the following spaces: a) span ((1, 2, 3), (2, 4, 6), (3, 6, 9))…

A: In this question, we will find the dimensions of the spaces.

Q: 9 Ex: Show that every finite space is compact space. Eulet die a trivial matrie on infinite catV(Y…

Q: let A be a set. A:=(a1,a2,a3) in R^3 such that a1 is greater than 0 but less than and equals to 1,…

Q: Suppose (X, d) is a metric space and E a non-empty subset of X. Prove that E is disconnected if and…

Question

Do the following task.
a) State formally the Heine-Borel theorem in R^k
b) Apply Heine-Borel theorem to show that if A is bounded, B is
closed such that B ⊆ A ⊆ R^k, then B is compact.
c) Give an example of set A and B in R such that A is bounded,
B is closed, B ⊆ A but A is not compact.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

Let (S,≼) be a linearly ordered set with the Least-Upper-Bound Property. Let A and B be non-empty and bounded below subsets of S. (i) Prove that A ∪ B is bounded below in S.(ii) Prove that inf(A ∪ B) ≼ inf(A) by using the definition of the infimum of a set in S.
Q1:- Let A, B, C, D, E be subsets of the Euclidean space R². Find their boundary, their interior, and their exterior. Conclude from here whether these sets are open, closed, or neither. a) A = {x € R² | d₂ (x,xo) ≤ 2}, where x₁ € R². b) B = Rx [a, b), where a, b E R, a < b. c) C = (a, b)² = (a, b) x (a, b), where a, b € R, a < b. d) D= (a) x [b, c), where a, b, c € R,b< c. e) E = {a} x {b, c), where a, b, c E R, b = c.
Prove disprove that (R,T co-finite) is Te-space or
Please help 6 and 7 qaution I upvote you
Let F subset of K be a finite extension. Then |Emb_F(K, L)| ≤ dim_F(K), where L is an algebraic closure of F containing K.
25) Let A be dense subset of (X, T) and let B be non-empty open subset of X then
let fiR> R the function defined in the fllowig farm fox=x²,indicate whether itisa continuous function Bases of Topology R which imeludes the family of all open sets using
Q1: (a) If (M, d) be a metric space and AC M. Prove that A is the largest open set in M that contained in A. then give an example to show that (A U B) # A* U B. (b) let (M, d)be a metric space and let A ≤ M. Then (AB) ≤ĀNB
1D
Show that Continnons bijection bijuction ut a compact topologin) space X unto on Hansdorft space Y is on homecomorph:3m₂ 13 n