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anonymous
 one year ago
can anybody teach me what balls are in metric space
anonymous
 one year ago
can anybody teach me what balls are in metric space

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Do 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.

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0Do you know a definition of an open ball / closed ball of radius r, around a point x, in R^n?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i know what a metric and metric spcace is

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0Ok, so you have a point x in R^n. The set of all points y, such that the distance xy<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 <=

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0please , what is a ball?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0please make me understand this concept

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0Did you read my explanation? Can you tell me what's the first point that didn't make sense?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0is the open ball of radius r, around x.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0first, what is a ball?

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0It'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.

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0dw: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.

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0So, 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?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0give examples please and solve. like\[[ B_1(2)] \]/

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0Ok, 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 x2 < 1 Can you solve this, and express the solution as an interval?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0hmm, no. it seems i will get values more than one or equal 1

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0dw:1442187702267:dw Well, here's a solution. If you have problems with this you should review high school algebra.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0understood. now ask me semilar questions

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0Ok, find the open ball of radius 5 around the point 3, on the real line.dw:1442187980593:dw

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0Ok. 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.

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0P(1,5) is a point in the plane.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh, this should be in \[R^2 \]?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0well, i cant solve that

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0That'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.

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0If you want to really understand these concepts you have to think about them in words as well. Not just 'solve equations'.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok but i can't sketch here

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0Can you use the drawing tool on the site?

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0Ok :) Anyway, you should do it on a paper, with different numbers then. Just to build your own intuition. Here's my solution

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0dw:1442188974052:dw

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0Can 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.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yes very good but can you solve that mathematically?

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0Yes, 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 <.

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.0Anyway, 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.
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