Are there an infinite number of open balls in an open set in a metric space? Let's start off by recalling the definition of an open set in a metric space:
A set $A$ in a metric space $(X,d)$ is open if for each point $x\in A$ there is a number $r\gt0$ such that $B_r(x)\subset A$
Where $B_r(x)$ denotes the open ball of radius $r$ at a point $x$,
$$B_r(x)=\{y\in X:d(x,y)\lt r\}$$
Supposing that our space is some form of the reals, $\mathbb R^n$, does this not mean that there are an infinite number of open balls, and hence an infinite number of points, within A?
This is my reasoning so far: take any point $x\in A$, and say that for some $r$ we have some $B_r(x)$ centered at $x$. Then, surely, there would be some point $x_1\in B_r(x)$ such that $d(x,x_1)\lt r$. But then, must not $x_1$ be in $A?$ And so would there not also exist some other open ball, $B_{r\,'}(x_1)$ with radius $r'$ centered at $x_1$, and then so on and so forth for the other points within that radius?

It is not clear to me what you are asking exactly.
Given a ball $B=B(x;r)$ in $\Bbb R^n$ all the balls $B(x;\rho)$ with $\rho<r$ are inside $B$ and so in whatever open set $A$ the ball $B$ is a subset of. This is entirely trivial.
A bit less trivial question would be if you can find infinitely many disjoint balls inside an open set $A\subseteq\Bbb R^n$, but I have no clue that this is your actual question.
