can anybody teach me what balls are in metric space

- anonymous

can anybody teach me what balls are in metric space

- chestercat

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- anonymous

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.

- beginnersmind

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

- anonymous

no

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- anonymous

i know what a metric and metric spcace is

- beginnersmind

Ok, so you have a point x in R^n. The set of all points y, such that the distance ||x-y||

- anonymous

please , what is a ball?

- anonymous

please make me understand this concept

- beginnersmind

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

- anonymous

is the open ball of radius r, around x.

- anonymous

first, what is a ball?

- beginnersmind

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.

- anonymous

ok

- beginnersmind

|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.

- beginnersmind

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?

- anonymous

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

- anonymous

i am with you

- beginnersmind

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?

- anonymous

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

- anonymous

|2-2|<1 only

- beginnersmind

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

- anonymous

understood. now ask me semilar questions

- beginnersmind

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

- anonymous

-2

- beginnersmind

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.

- beginnersmind

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

- anonymous

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

- beginnersmind

Yes.

- anonymous

well, i cant solve that

- beginnersmind

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.

- beginnersmind

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

- anonymous

ok but i can't sketch here

- beginnersmind

Can you use the drawing tool on the site?

- anonymous

no

- beginnersmind

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

- beginnersmind

|dw:1442188974052:dw|

- beginnersmind

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.

- anonymous

yes very good but can you solve that mathematically?

- beginnersmind

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 <.

- beginnersmind

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.

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