Suppose that X satisfies the Bolzano-Weierstrass Property and that A and B are compact subsets of X.
Question:
dist (A, B) := inf{p(x, y) : x ∈ A and y ∈ B},
then dist (A, B) > 0. Show that even in the space R2, there exist subsets A and B which are closed and satisfy A ∩ B = θ, but dist(A, B) = 0.
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Answer rating: 91% (12 reviews)
a Since both sets are nonempty and px y is bounded ...View the full answer
Answered By
Shristi Singh
A freshman year metallurgy and material science student in India.
4.80+
2+ Reviews
10+ Question Solved
Related Book For
Question Posted:
Students also viewed these Numerical Analysis questions
-
Let X be a metric space. a) Prove that if E X is compact, then E is sequentially compact (see Exercise 10.1.10). b) Prove that if X is separable and satisfies the Bolzano-Weierstrass Property, then...
-
Suppose that X is a separable metric space which satisfies the Bolzano-Weierstrass Property, that Y is a complete metric space, and that E is a bounded subset of X. Prove that a function f: E Y is...
-
Prove that the Bolzano-Weierstrass Property does not hold for C[a, b] and ||f|| (see Example 10.6). Namely, prove that if fn(x) = xn then ||fn|| is bounded but ||fnk - f|| does not converge for any f...
-
Various statistical software packages tend to provide different measures of association. Use statistical software to calculate the Goodman-Kruskal gamma, Somers D, or Kendalls tau-a for the space...
-
A-Rod Fishing Supplies had sales of $2,500,000 and cost of goods sold of $1,710,000. Selling and administrative expenses represented 10 percent of sales. Depreciation was 6 percent of the total...
-
Clean Pool Service had the following selected accounts and normal balances listed on its December 31 adjusted trial balance: Service Revenue Accounts Payable Salaries Expense Common Stock Utilities...
-
With reference to Exercise 11.65, test the null hypothesis \(\beta=1.5\) against the alternative hypothesis \(\beta>1.5\) at the 0.01 level of significance. Data From Exercise 11.65 11.65 The data...
-
On June 1, High Performance Cell Phones sold $19,000 of merchandise to Andrew Trucking Company on account. Andrew fell on hard times and on July 15 paid only $7,000 of the account receivable. After...
-
A cooperative group housing society wants to establish a sinking fund to accumulate Rs.25,00,000 in 3 years to repair the society fencing. The fund earns 12% interest compounded monthly. If there are...
-
Suppose a private bank client has US$1,000,000 to investment. Advise the client on a derivatives strategy that take advantage of the current market opportunity. The components of your trading...
-
Prove that if V is open in a separable metric space X, then there are open balls B1, B2,... such that Prove that every open set in R is a countable union of open intervals. jEN
-
a) Prove that Cantor's Intersection Theorem holds for nested compact sets in an arbitrary metric space; that is, if H1, H2, ... is a nested sequence of nonempty compact sets in X, then b) Prove that...
-
Listed in the tables are six springs described in customary units and five springs described in SI units. Investigate these squared-and-ground-ended helical compression springs to see if they are...
-
The list of 3 jobs Ross Co. worked on during the year: Cost Job B1 Job O1 Job B2 Prime costs $4,100 $6,250 $7,400 Manufacturing overhead ? ? ? Bob applies overhead at a rate of 65% of prime costs....
-
The client is employed in Amazon, India, and receives a competitive annual package that includes ESOPs that have increased in value. With a taste for success, your client is investing a large amount...
-
Write a C program named as countD.c that takes input text file and a digit as arguments, outputs how many times the digit appears in the file. If the text file does not exist, print out an error...
-
Economic progress within a country seems to depend on how well property rights are protected. Explain the relationship between property rights, corruption, and economic progress and their importance...
-
Discuss the characteristics of the rule of law, according to Hayek.
-
In Exercises find the extrema and the points of inflection (if any exist) of the function. Use a graphing utility to graph the function and confirm your results. f(x) = 2+ ex(4- 2x)
-
What are technical skills At what level are they most important and why?
-
Consider the etch rate data in Exercise 10-21. Suppose that if one population variance is twice as large as the other, we want to detect this with probability at least 0.90 (using a = 0.05). Are the...
-
Consider the diameter data in Exercise 10-17. Construct the following: (a) A 90% two-sided confidence interval on 1/ 2 (b) A 95% two-sided confidence interval on 1/ 2. Comment on the comparison of...
-
Consider the foam data in Exercise 10-18. Construct the following: (a) A 90% two-sided confidence interval on 2/1 2/2 (b) A 95% two-sided confidence interval on 2/1 2/2. Comment on the comparison of...
-
1. Electric potential is given by V=6x-8xy-8y+6yz - 4z Then magnitude of electric force acting on 2C point charge placed on origin will be :- (1) 2N (2) 6N (3) 8N (3) 8N (4) 20 N 2. Figure shows...
-
23. A convex mirror used for rear view on an automobile has a radius of curvature of 3m. If a bus is located at 5m from this mirror, find the position, nature and magnification of the image. 24. An...
-
17. A concave mirror produces a real image 10mm tall, of an object 2.5mm tall placed at 5cm from the mirror. Calculate focal length of the mirror and the position of the image. 18. An object is...
Study smarter with the SolutionInn App