Finding the sphere?

Question

(Discrete metric) Let $X$ be a nonempty set. Define a map on $ X \times X$ by $d(x,y) = \begin{cases} 0 &\text{if } x = y \\ 1 &\text{ if } x\neq y. \end{cases}$

Let $r > 0$ and let $x \in X $. Find the sphere $S(x,r)$.

My attempt: I know that $\{x\} \subseteq S(x,r)$.

Any hints/solution will be appreciated.

Thank you.

Answer 1

You have $S(x,r) = \{ y \in X \mid d(x,y) = r \}$. Then $S(x,1) = X \setminus \{ x \}$ and $S(x,r) = \emptyset$ for $r >0, r \ne 1$.