Question 4

Question

What is the ball topology on a metric space?

Answer

A ball has two types, either an open (denoted by $B_r(x)$) or closed ball (denoted by $B_r[x]$).

Let $d$ be a metric, on a set $X$, giving us a metric space $(X,d)$.

And a ball has two properties:

• A radius $r>0$
• Centre point $x\in X$.
  Then the ball is defined as a set of points in $X$ that is of distance less than $r$ away from $x$
  $B_r(x)=\{y\in X|d(y,x)<r\}.$

Closed Ball

Note

I assume this is not required as I don’t think this was covered in the lecture.

A closed ball is similar to an open ball. However includes the points on the boundary.

It has similar properties as the open ball, but it is defined as the points less than or equal to $r$ away from $p$
$B_r[x]=\{y\in X|d(y,x)\le r\}.$

In a nutshell

Replaces the closed one replaces the $<$ with $\le$ in $d(x,y)<r$.