Can an open ball be closed if the open ball contains infinite points? Consider a metric space $(X,d)$. Is the following statement true?
An non-finite open ball in X with finite radius is never closed. Non-finite in this sense means that the open ball contains an infinite amount of points.

When $d$ is a metric then $e_1=d/(1+d)$ and $e_2=\max (d,1)$ are metrics equivalent to $d,$ that is, they generate the same topology that $d$ does. Since $\sup e_1\leq 1\geq \sup e_2,$ an open $e_1$-ball or   $e_2$-ball of radius $2$ is the whole space, which is both open and closed, regardless of whether it is a finite or infinite space.
Another common misconception is that $\{y:d(y,x)\leq r\}$ is always the closure of $\{y:d(y,x)<r\}.$
