Please solve number 5 1. (X, d) is a metric space and A and B are subsets of X. Prove eachof the following if the boundary of a set A, denoted as bd(A), is definedas bd(A) = An (X - A). Prove each of the following:(i) bd(A) is a closed set.(ii) A set A is closed if and only if bd (A) CA.(iii) clA = Aº U bd.A.(iv) If the metric d on X is the discrete metric, discuss the boundaryof a set AC X.3. If G is an open dense subset of X, then X - G is nowhere dense,where G is a non-empty proper subset of a metric space X.4. Let U, V be subsets of a metric space X and V be open. Show thatif Un V = 0, then clUnV = 0."5. Give an example of a family {Un new} of open subsets of Rwith usual metric, such that new Un is not open.

Answered: Image /qna-images/answer/d9652c33-6e6c-4ca7-a05f-7569238bee6c.jpg

Answered: 5. Give an example of a family {Un new}…

Math

Advanced Math

5. Give an example of a family {Un new} of open subsets of R, with usual metric, such that new Un is not open.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 20E: Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not...

See similar textbooks

Related questions

Q: 8-3) show that T is a linear transformation by finding a matrix that implements the mapping. Note…

A: (a) The given linear transformation is Tx1,x2=2x2-3x1,x1-4x2,0,x2 (b) The given linear…

Q: Find the Laplace transform F(s) = L{f(t)} of the function f(t) = 8th(t - 7), defined on the interval…

Q: 2. Prove the following: For any simple graph G = (V. E), V can be partitioned into X, Y, Z such that…

A: The objective is to prove that for any simple graph G=V,E, EX,Y,Z≥2E3. A simple graph is a graph…

Q: A conservative vector field is given by: F = (-z - 1)i + (-2y + 2)j + (-x - 1)k What 2 options…

Q: Write and simplify the integral that gives the arc length of the following curves on the given…

A: Given: A curve y=lnx between x=1 and x=b. To Find: Integral to evaluate the arc length using some…

Q: David invested $2,100 at the beginning of every 6 months in an RRSP for 11 years. For the first 5…

A: Given investment amount PMT=$ 2100Interest rate r=3.80%=3.80100=0.038Compounded semi annually m=2…

Q: 12. : R³ → R², 7(e₁) = (1,3), 7(e₂) = (4,2), and 7(e) = (-5, 4), where e₁,e2, e3 are the columns of…

Q: A candy company has 195 kg of chocolate-covered nuts and 105 kg of chocolate-covered raisins to be…

A: Let, x: Kg of the half nuts and half raisins misture, and y: Kg of the 3/4 nuts and 1/4 raisins…

Q: The matrix 6 [14 - 9 -4 k is invertible if and only if k

Q: Is the following linear transformation injective? Surjective? (Justify both with full sentences). T…

A: The given linear transformation Tx, y=x+3y, y-2x, y. We have to check T is injective, or surjective.

Q: Inference Accept the following Premises 1. 2. 3. If fish can fly, birds can Swim If birds can swim,…

A: We have been given: 1. If fish can fly, birds can swim. 2. If birds can swim, squirrels can talk. 3.…

Q: 1. Let R be a commutative ring with unity a. Let ✅ be a collection of ideals in R. Prove that ea is…

Q: c.) 3x²y"-6xy³y' + 4x = 0 d.) 3xd³y_d²y dx³ dx² -5xy = 3x²y

Q: Let P(x) be given by "x is a prime number," where the domain of x is the set of all positive…

Q: 4-1) describe the possible echelon forms of the matrix. (a) A is a 3x3 matrix with linearly…

A: Given: In this question, the given details are, 4-1) Here, to describe the possible echelon forms of…

Q: Inference Accept the following Premises 1. If fish can fly, birds can Swim If birds can swim,…

A: Given: 1. If fish can fly, birds can swim. 2. If birds can swim, squirrels can talk. 3. If birds can…

Q: 1. Select the values that are not solutions. x > 7 2685

A: The given inequality x>7. We have to find the values of x that are not solutions.

Q: Perform the following congruence computations. Make sure that the number you enter is > 0 and ≤ N-1,…

Q: Find the equation of the line which is at a perpendicular distance of 5 units from the origin and…

Q: Calculate Vf using suffix notation and the chain rule: ƒ(r) = 3(a ⋅r)² − a²r² (a is a vector…

Q: Whether the polynomial x² + 2x + 2 over Z3, Z5, and Z7 is irreducible or not and state all zeros.

Q: C. Solve the initial value problem of the following DE using Laplace Transform. b.) y"" - 3y" = te³t…

Q: 2 proofs of binomial identities applied to Pell-Lucas numbers.

A: The Pell-Lucas numbers are a sequence of integers defined by the recurrence relation: P(n) = 2P(n-1)…

Q: Solve the partial differential equation (D^2-DD'-2D'^2)z=x^3 Where D=z_x , D'= z_y

A: Given partial differential equation is D2-DD'-2D'2z=x3 where D≡∂z∂x and…

Q: Giving f(0.23) = 0.362, f(0.53) = 0.143, f(0.86) = −1.143, please use inverse interpolating method…

A: Given that f0.23=0.362, f0.53=0.143, f0.86=-1.143. Using inverse interpolating method to get the…

Q: (a) For the following example, U = with a 0 and d‡0. Find x 0 such that a 0 0 b d 0 Ux=0 C 0 (b)…

Q: Find the general solution of the given second-order differential equation. y" - y' - 12y = 0 y(x) =

Q: 37. Asymptote Behavior How close to -3 do we have to take x so that > 10,000 (x + 3)?

A: We can solve this inequality by isolating x and finding the values of x that make the inequality…

Q: Find the equation of the line passing through the point (2,2) and cutting off intercepts on the axes…

A: The equation of a line in the intercept form is xa+yb=1this line passes through…

Q: Section 1.7 Problem 3. Mark each statement True or False, and justify your answer. (T/F) The columns…

A: Disclaimer: Since you posted multiple questions, we will provide the solution only to the first five…

Q: Find the equation of an ellipsoid passing through the points (±4,0,0), (0, 18,0) and (0, 0, ±14).…

A: The equation of the ellipsoid is:x2a2+y2b2+z2c2=1 We have given the ellipsoid passing through the…

Q: 3. f(x, y) = x² x+y + y²-1 -; (1, 1), (-1, 1).

Q: Let F: R³ → R4 be the linear transformation that satisfies 2 2 1 -0-0-0-0-0-0 F1 = 2 F = 1 1 -2 1 F1…

Q: -D- (a) For what values of h is v3 in Span {V1, V2}? Justify your answer. Problem 3. Let V₁ = 3,V₂ =…

A: We will use basic definition of linear span of two vectors. We are given here three vectors; which…

Q: 1 lim h-0 00 04 O-1 01 ODNE O None of the above 1 (5 (2) √5²) 5 00 01 2 18116 7 30 +h) - 2 + 0′(1).…

Q: 3. Use method of undetermined coefficients 2X y" - Ay 8xe

Q: A (В C The continuous function f is known to be increasing for all x. Selected values of f are given…

Q: Three regions are defined in the figure.

A: Given that three regions are defined in the figure: . To find the…

Q: Calculate ffs f(x, y, z) dS For x² + y² = 4, 0≤ z ≤ 4; Is f(x, y, z) ds = k f(x, y, z)= ez

A: We can find surface integral.

Q: 31. (a) How many different spanning trees does the network shown in Fig. 7-32(a) have? (b) How many…

Q: B. Find the inverse Laplace transform using the following methods: 1. Using Partial Fraction…

A: Given Data: Let us consider the given data, Fs=4s+5s-12s+3. To Find: The inverse laplace transform…

Q: Show that the application of L'hopital's rule is relevant for lim (+). X-1

Q: b~: d bad b d bed d~:: b~ b=d d :: b~ b=d

Q: Find the form of yo to use in the method of undetermined coefficients v+5V tây e + xe a. Ae* + Bxe²x…

Q: DIRECTION: Use the formula below to solve for the derivative of the following. Show the complete…

Q: In Exercises 2.4.5 and 2.4.6 find all solutions. Also, plot a direction field and some integral…

Q: #20] Show that the subsets of the plane ( = {cx₁y) √ 2 < x ³² + y ² <≤ 4} ane of

A: Solution:- Given set C={(x,y)| 2<x^2+y^2<4 } We have to show this is open

Q: Arc length by calculator a. Write and simplify the integral that gives the arc length of the fol-…

A: Given : The equation of the given curve is y=8x2. To Find : a) Write and simplify the integral that…

Q: Using all three conditions of continuity, determine if the following functions are continuous at x =…

A: We can determine whether the function is continuous or not.

Q: B. Find the inverse Laplace transform using the following methods: I. Using Partial Fraction…

A: Sol:- L-16s+17s2+8s+17Expand 6s+17s2+8s+17: 6·s+4s+42+1-7·1s+42+1=L-16·s+4s+42+1-7·1s+42+1Use the…

Question

Please solve number 5

1. (X, d) is a metric space and A and B are subsets of X. Prove each
of the following if the boundary of a set A, denoted as bd(A), is defined
as bd(A) = An (X - A). Prove each of the following:
(i) bd(A) is a closed set.
(ii) A set A is closed if and only if bd (A) CA.
(iii) clA = Aº U bd.A.
(iv) If the metric d on X is the discrete metric, discuss the boundary
of a set AC X.
3. If G is an open dense subset of X, then X - G is nowhere dense,
where G is a non-empty proper subset of a metric space X.
4. Let U, V be subsets of a metric space X and V be open. Show that
if Un V = 0, then clUnV = 0.
"
5. Give an example of a family {Un new} of open subsets of R
with usual metric, such that new Un is not open.

Expert Solution

This question has been solved!

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

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here