Topology
2nd Edition
ISBN: 9780134689517
Author: Munkres, James R.
Publisher: Pearson,
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Chapter 4.30, Problem 6E
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Chapter 4 Solutions
Topology
Ch. 4.30 - Show that l and I02 are not metrizable.Ch. 4.30 - Which of our four countability axioms does S...Ch. 4.30 - Which of our four countability axioms does in the...Ch. 4.30 - Let A be a closed subspace of X. Show that if X is...Ch. 4.30 - Prob. 10ECh. 4.30 - Let f:XY be continuous. Show that if X is...Ch. 4.30 - Let f:XY be continuous open map. Show that if X...Ch. 4.30 - Show that if X has a countable dense subset, every...Ch. 4.30 - Show that if X is Lindelof and Y is compact, then...Ch. 4.30 - Give I the uniform metric, where I=[0,1]. Let...
Ch. 4.30 - Prob. 16ECh. 4.30 - Prob. 17ECh. 4.30 - Prob. 18ECh. 4.31 - Show that if X is regular, every pair of points of...Ch. 4.31 - Show that if X is normal, every pair of disjoint...Ch. 4.31 - Show that every order topology is regular.Ch. 4.31 - Prob. 4ECh. 4.31 - Prob. 5ECh. 4.32 - Which of the following spaces are completely...Ch. 4.32 - Prob. 8ECh. 4.32 - Prove the following: Theorem: If J is uncountable,...Ch. 4.32 - Prob. 10ECh. 4.33 - Examine the proof of the Urysohn lemma, and show...Ch. 4.33 - a Show that a connected normal space having more...Ch. 4.33 - Give a direct proof of the Urysohn lemma for a...Ch. 4.33 - Prob. 4ECh. 4.33 - Prob. 5ECh. 4.33 - Prob. 8ECh. 4.34 - Give an example showing that a Hausdorff space...Ch. 4.34 - Give an example showing that a space can be...Ch. 4.34 - Let X be a compact Hausdorff space. Show that X is...Ch. 4.34 - Let X be a locally compact Hausdorff space. Is it...Ch. 4.34 - Let X be a locally compact Hausdorff space. Let Y...Ch. 4.34 - Check the details of the proof of Theorem 34.2.Ch. 4.34 - A space X is locally metrizable if each point x of...Ch. 4.34 - Show that a regular Lindelof space is metrizable...Ch. 4.35 - Show that the Tietze extension theorem implies the...Ch. 4.35 - In the proof of the Tietze theorem, how essential...Ch. 4.35 - Let X be metrizable. Show that the following are...Ch. 4.35 - Let Z be a topological space. If Y is a subspace...Ch. 4.35 - Prob. 5ECh. 4.35 - Let Y be a normal space. The Y is said to be an...Ch. 4.35 - a Show the logarithmic spiral...Ch. 4.35 - Prove the following: Theorem. Let Y be a normal...Ch. 4.36 - Prove that every manifold is regular and hence...Ch. 4.36 - Let X be a compact Hausdorff space. Suppose that...Ch. 4.36 - Let X be a Hausdorff space such that each point of...Ch. 4.36 - Prob. 5ECh. 4.SE - Consider the following properties a space may...Ch. 4.SE - Consider the following properties a space may...Ch. 4.SE - Prob. 3SECh. 4.SE - Consider the following properties a space may...
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- Select the best statement. A. If a set of vectors includes the zero vector 0, then the set of vectors can span R^ as long as the other vectors are distinct. n B. If a set of vectors includes the zero vector 0, then the set of vectors spans R precisely when the set with 0 excluded spans Rª. ○ C. If a set of vectors includes the zero vector 0, then the set of vectors can span Rn as long as it contains n vectors. ○ D. If a set of vectors includes the zero vector 0, then there is no reasonable way to determine if the set of vectors spans Rn. E. If a set of vectors includes the zero vector 0, then the set of vectors cannot span Rn. F. none of the abovearrow_forwardWhich of the following sets of vectors are linearly independent? (Check the boxes for linearly independent sets.) ☐ A. { 7 4 3 13 -9 8 -17 7 ☐ B. 0 -8 3 ☐ C. 0 ☐ D. -5 ☐ E. 3 ☐ F. 4 THarrow_forward3 and = 5 3 ---8--8--8 Let = 3 U2 = 1 Select all of the vectors that are in the span of {u₁, u2, u3}. (Check every statement that is correct.) 3 ☐ A. The vector 3 is in the span. -1 3 ☐ B. The vector -5 75°1 is in the span. ГОЛ ☐ C. The vector 0 is in the span. 3 -4 is in the span. OD. The vector 0 3 ☐ E. All vectors in R³ are in the span. 3 F. The vector 9 -4 5 3 is in the span. 0 ☐ G. We cannot tell which vectors are i the span.arrow_forward
- (20 p) 1. Find a particular solution satisfying the given initial conditions for the third-order homogeneous linear equation given below. (See Section 5.2 in your textbook if you need a review of the subject.) y(3)+2y"-y-2y = 0; y(0) = 1, y'(0) = 2, y"(0) = 0; y₁ = e*, y2 = e¯x, y3 = e−2x (20 p) 2. Find a particular solution satisfying the given initial conditions for the second-order nonhomogeneous linear equation given below. (See Section 5.2 in your textbook if you need a review of the subject.) y"-2y-3y = 6; y(0) = 3, y'(0) = 11 yc = c₁ex + c2e³x; yp = −2 (60 p) 3. Find the general, and if possible, particular solutions of the linear systems of differential equations given below using the eigenvalue-eigenvector method. (See Section 7.3 in your textbook if you need a review of the subject.) = a) x 4x1 + x2, x2 = 6x1-x2 b) x=6x17x2, x2 = x1-2x2 c) x = 9x1+5x2, x2 = −6x1-2x2; x1(0) = 1, x2(0)=0arrow_forwardFind the perimeter and areaarrow_forwardAssume {u1, U2, us} spans R³. Select the best statement. A. {U1, U2, us, u4} spans R³ unless u is the zero vector. B. {U1, U2, us, u4} always spans R³. C. {U1, U2, us, u4} spans R³ unless u is a scalar multiple of another vector in the set. D. We do not have sufficient information to determine if {u₁, u2, 43, 114} spans R³. OE. {U1, U2, 3, 4} never spans R³. F. none of the abovearrow_forward
- Assume {u1, U2, 13, 14} spans R³. Select the best statement. A. {U1, U2, u3} never spans R³ since it is a proper subset of a spanning set. B. {U1, U2, u3} spans R³ unless one of the vectors is the zero vector. C. {u1, U2, us} spans R³ unless one of the vectors is a scalar multiple of another vector in the set. D. {U1, U2, us} always spans R³. E. {U1, U2, u3} may, but does not have to, span R³. F. none of the abovearrow_forwardLet H = span {u, v}. For each of the following sets of vectors determine whether H is a line or a plane. Select an Answer u = 3 1. -10 8-8 -2 ,v= 5 Select an Answer -2 u = 3 4 2. + 9 ,v= 6arrow_forwardSolve for the matrix X: X (2 7³) x + ( 2 ) - (112) 6 14 8arrow_forward
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