Let $$X$$ be a metric space, let $$x \in X$$ and let $$\varepsilon > 0$$ be a real. The open ε-ball around $$x$$ (or in more detail, the open ball around $$x$$ of radius $$\varepsilon$$) is the subset of $$X$$ given by
[ $$B(x, \varepsilon) = \{y \in X : \text{what condition?} \}$$ ]
Similarly, the closed ε-ball around $$x$$ is
[$$\bar{B}(x, \varepsilon) = \{y \in X : \text{what condition?} \}$$ ]