In

algebraic topology, an

**open ball** centered on the point

`p` is a set of some points

`x` in a

metric space X, such that the

distance between

`x` and

`p` is less than some constant r, the '

radius' of the ball. This is usually written

B_{X}(`p`, r)

and is equivalent to the set:

{`x` in X : `d`(`x`, `p`) < r}

(where

`d` is the

distance function of the metric space, telling us the distance between its two arguments in that space.)

The construction may be used in order to define an open set as a subset `V` of X where for any point in it there is an open ball with nonzero and positive r, which is a subset of `V`.

That is

for all `v` in `V`, there exists r > 0 such that B_{X}(`v`, r) is a subset of `V`.

It also comes in handy for defining

continuous functions between metric spaces. A function

`f` mapping from

`X` to

`Y`, where both

`X` and

`Y` are metric spaces, is continuous at some point

`x` in

`X` only in the case that for every

`e` > 0 there is some

`d` > 0 such that for any member

`b` of B

_{X}(

`x`,

`d`),

`f`(

`b`) is in B

_{Y}(

`f`(

`x`),

`e`).