Question

In: Advanced Math

Two sets are separated if the intersection of the closure of one of the sets with...

Two sets are separated if the intersection of the closure of one of the sets with the other is empty.

a) Prove that two closed and disjoint sets of some metric space are separate.

b) Prove that two open and disjoint sets of some metric space are separate.

Solutions

Expert Solution

Check image for complete answer.

You just need the definition of closed and open sets.

For closed set A , we have Closure of A equals A.

For open set A , we have for each x in A there exists open neighborhood N which contains x and is completely contained in A.

x is in closure of A *iff* for each open neighborhood N containing x we have N intersection A is empty.

Colby Messinger answered 8 months ago
Next > < Previous

Related Solutions

The intersection (∩) of two sets (s1, s2) is the set of all elements that are...
The intersection (∩) of two sets (s1, s2) is the set of all elements that are in s1 and are also in s2. Write a function (intersect) that takes two lists as input (you can assume they have no duplicate elements), and returns the intersection of those two sets (as a list) without using the in operator or any built-in functions, except for range() and len(). Write some code to test your function, as well. Note: Do not use the...
Prove that the intersection of two compact sets is compact, using criterion (2).
  Question: Prove that the intersection of two compact sets is compact, using criterion (2). Prove that the intersection of two compact sets is compact, using criterion (1). Prove that the intersection of two compact sets is compact, using criterion (3). Probably the most important new idea you'll encounter in real analysis is the concept of compactness. It's the compactness of [a, b] that makes a continuous function reach its maximum and that makes the Riemann in- tegral exist. For...
Use MATLAB to determine the intersection point for the sets of equations shown below.
Use MATLAB to determine the intersection point for the sets of equations shown below. Use the backslash when possible.Graph the equations with different colors, and plot the intersection point with a red star marker. If the equations do not intersect, graph them anyway. Include a legend: a) -4x1 + 2x2 = 10                                                                                b)-4x1+2x2^3 = 10 x1 + x2 = -8                                                                                           x1+ x2 = -8                   
1)Prove that the intersection of an arbitrary collection of closed sets is closed. 2)Prove that the...
1)Prove that the intersection of an arbitrary collection of closed sets is closed. 2)Prove that the union of a finite collection of closed sets is closed
Assume the following happened: One of the stop signs at the intersection of two Coquitlam roads...
Assume the following happened: One of the stop signs at the intersection of two Coquitlam roads is knocked over on Saturday evening; The city Roads & Transportation Service (responsible for “signs down, missing or vandalized”) gets a call to their Engineering Customer Service very early on Sunday morning (so now they know the sign is down) but decides to wait until Monday morning to replace the sign, in order to save $1,250 overtime pay to its road crew; Sunday evening,...
Markets are separated into two broad categories based on their level of competition. On the one...
Markets are separated into two broad categories based on their level of competition. On the one hand there are perfectly competitive markets and on the other hand, there are all other market structures which are grouped under imperfect competition. Markets under imperfect competition are generally described by fewer firms with each firm having some degree of market power depending on the number of firms in the market. Discuss the fundamental differences between perfect competition and imperfect competition (include a discussion...
Markets are separated into two broad categories based on their level of competition. On the one...
Markets are separated into two broad categories based on their level of competition. On the one hand there are perfectly competitive markets and on the other hand, there are all other market structures which are grouped under imperfect competition. Markets under imperfect competition are generally described by fewer firms with each firm having some degree of market power depending on the number of firms in the market. Discuss the fundamental differences between perfect competition and imperfect competition (include a discussion...
Markets are separated into two broad categories based on their level of competition. On the one...
Markets are separated into two broad categories based on their level of competition. On the one hand there are perfectly competitive markets and on the other hand, there are all other market structures which are grouped under imperfect competition. Markets under imperfect competition are generally described by fewer firms with each firm having some degree of market power depending on the number of firms in the market. Discuss the fundamental differences between perfect competition and imperfect competition (include a discussion...
Two automobiles with equal mass approach an intersection.  One vehicle is traveling eastward and the other...
Two automobiles with equal mass approach an intersection.  One vehicle is traveling eastward and the other vehicle is traveling northward.  The vehicles collide and stick together. After impact they travel on the road and then travel on a gravel covered shoulder until both cars stop. Only the northward bound vehicle left skid marks at an angle of 55 degrees north of east.  A police officer posted at the intersection clocked the eastward vehicle at 29 mph before impact. The northward...
Two automobiles with equal mass approach an intersection. One vehicle is traveling eastward and the other...
Two automobiles with equal mass approach an intersection. One vehicle is traveling eastward and the other vehicle is traveling northward. The vehicles collide and stick together. After impact they travel on the road and then travel on a gravel covered shoulder until both cars stop. Only the northward bound vehicle left skid marks at an angle of 55 degrees north of east. A police officer posted at the intersection clocked the eastward vehicle at 29 mph before impact. The northward...
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT