How can I explain to my professor his argument invokes the AC? This is not the standard definition, but my topology professor restricted contexts in metric spaces.

Definition
An open set $U$ in a metric space $X$ is a subset of $X$ such that the interior of it and $U$ Itself are identical. (Interior point $x$ of a subset $A$ is a point of which there exists a ball containing $x$, contained in $A$.

With this definition, He showed that "Every open set in a metric space is a union of balls."
Below is his argument:

Assume that $U$ is open.
Let $x\in U$.
Then, by definition, there exists $r(x)$ such that $B(x,r(x))\subset U$, so that $U=\bigcup_{x\in U} B(x,r(x))$.

I'm not asking that this can be resolved easily without Axiom of Choice, but I'm asking how do I explain this to my professor his argument invokes Axiom of Choice. I told him that choosing $r(x)$ for each $x$ is a consequence of the axiom of choice, but he does not seem to understand me. Please help.

Setting aside the question of whether his argument uses the axiom of choice, the result doesn’t depend on it, and the proof is basically the same. Define $r(x)=\frac{1}{n}$, where $n$ is the smallest positive integer for which $B(x,\frac{1}{n})\subseteq U$.
