If $a$ is an arbitrary element of $X$, then what do the balls of radius $1/2$ and $1$ and $2$ around $a$ contain?

Question

If $X$ is any non-empty set, then the function

$d(x,y) = \begin{cases} 0, & x = y \\ 1, & otherwise. \end{cases}$

is a metric, known as the discrete metric.

If $a$ is an arbitrary element of $X$, then what do the balls of radius $1/2$ and $1$ and $2$ around $a$ contain?

I have absolutely no idea how to go about answering such a question. I understand what a metric is, I understand what the discrete metric is, and I understand the concept of open balls. However, it is not at all clear to me how one would find what the balls around some arbitrary point $a$ would contain.

In the same section as this problem, I did come across something about finding unique limits and disjoint balls, but that seems to be unrelated to the question at hand? And so, I cannot think of how to proceed with solving this problem.

I would greatly appreciate it if people could please take the time to explain this.

Answer 1

You just interpret literally. Assuming "ball" means "open ball" here (which is pretty standard) a ball of radius r around a just means everything with a distance less than r from a. Since any point b other than a has distance 1 from a, the balls of radius 1/2 and of radius 1 are just $\{a\}$ and the ball of radius 2 around a is the entire space $X$.