Is a ball that is not closed necessarily open? I understand that we can have an interval that is neither closed or open, such as [0,1), but can a ball be neither closed or open? If so, what is an example of this?

A ball is always either closed or open. In fact, there are only two types of balls.


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*Open balls: sets of the form $\{y\in X: d(x,y)<r\}$.

*Closed balls: sets of the form $\{y \in X: d(x,y)\le r\}$.
Since, for a fixed $x$, $y \mapsto d(x,y)$ is continuous, it follows from the definitions that open balls are open and that closed balls are closed.
