A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • one year ago

can anybody teach me what balls are in metric space

  • This Question is Closed
  1. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Do you know what a metric space is? If you do, then metric balls (or open balls) are a concept to study sets, they help you a lot in topology and measure theory in order to verify or falsify if a set is open/closed/clopen.

  2. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Do you know a definition of an open ball / closed ball of radius r, around a point x, in R^n?

  3. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    no

  4. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i know what a metric and metric spcace is

  5. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ok, so you have a point x in R^n. The set of all points y, such that the distance ||x-y||<r, is the open ball of radius r, around x. You have the same thing for metric spaces, by replacing the euclidean distance to the distance given by the metric. So it's the set of all point y such that d(x,y)<r. Closed balls work the same, but you replace < by <=

  6. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    please , what is a ball?

  7. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    please make me understand this concept

  8. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Did you read my explanation? Can you tell me what's the first point that didn't make sense?

  9. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is the open ball of radius r, around x.

  10. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    first, what is a ball?

  11. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    It's a set of points, that are within a given distance from a 'center'. For example the set of all real numbers in the interval ]-1,1[ is a ball of radius 1, centered around the point 0.

  12. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok

  13. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1442186744374:dw| In one dimension. |dw:1442186814103:dw| In two dimensions. The ball is the set of points inside the ball, because they are all closer to the center than r.

  14. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    So, to describe a ball, you need a center and a distance, r. The points that are closer to the center than r, are in the ball. There's two kinds of balls: open balls require the distance to be strictly less then r. closed balls allow equality as well. For the two dimensional case an open ball would be the disk of a circle without the perimeter. The closed ball would be the disk plus the parameter. With me so far?

  15. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    give examples please and solve. like\[[ B_1(2)] \]/

  16. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i am with you

  17. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ok, so this is an example in R. The center of the ball is at 2 and the radius is 1. So we are looking for the set of all numbers, where |x-2| < 1 Can you solve this, and express the solution as an interval?

  18. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    hmm, no. it seems i will get values more than one or equal 1

  19. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |2-2|<1 only

  20. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1442187702267:dw| Well, here's a solution. If you have problems with this you should review high school algebra.

  21. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    understood. now ask me semilar questions

  22. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ok, find the open ball of radius 5 around the point 3, on the real line.|dw:1442187980593:dw|

  23. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    -2<x<8

  24. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ok. Here's a difficult example. Let P = (-1,-5) and r = 2. Describe the open ball \(B_2(P)) \) in words and give the inequality the coordinates of any point Q(x,y) should satisfy to be an element of the ball.

  25. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    P(-1,-5) is a point in the plane.

  26. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh, this should be in \[R^2 \]?

  27. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yes.

  28. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    well, i cant solve that

  29. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    That's ok. Remember, the ball is the set of all points that are less than a distance 2, from the point (-1,-5). Sketch the point on a coordinate system and see if you can 'guess' which points are closer than a distance of 2 from it.

  30. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    If you want to really understand these concepts you have to think about them in words as well. Not just 'solve equations'.

  31. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok but i can't sketch here

  32. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Can you use the drawing tool on the site?

  33. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    no

  34. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ok :) Anyway, you should do it on a paper, with different numbers then. Just to build your own intuition. Here's my solution

  35. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1442188974052:dw|

  36. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Can you see my drawing though? It's supposed to be a circle of radius 2 centered on (-1,-5). I used a dashed line to indicate that the perimeter isn't included.

  37. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes very good but can you solve that mathematically?

  38. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yes, if you need to. You write the equation of a circle with center P(-1,-5) and radius r=2. Then instead of = you write <.

  39. beginnersmind
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Anyway, think I lost you again. The best advice I can give is that you think about what distance means in \(\Bbb R,\Bbb R^2 \text{ and } \Bbb R^n \), and how it relates to stuff like absolute values. Metric spaces generalize distance. But you need to understand one particular example really well to be able to deal with the general concept.

  40. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.