Question

can anybody teach me what balls are in metric space

Answer 1

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.

Answer 2

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

Answer 3

no

Answer 4

i know what a metric and metric spcace is

Answer 5

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 <=

Answer 6

please , what is a ball?

Answer 7

please make me understand this concept

Answer 8

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

Answer 9

is the open ball of radius r, around x.

Answer 10

first, what is a ball?

Answer 11

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.

Answer 12

ok

Answer 13

In one dimension.

In two dimensions.

The ball is the set of points inside the ball, because they are all closer to the center than r.