I noticed today that an argument used in a classical result from topology was not clear to me.
The framework is the following : let $A$ be an open set of a metric space $X$, for all $x\in A$ there exists $\epsilon_x>0$ such that the ball $B(x,\epsilon_x)$ is included in $A$.
Then the argument annoying me is the following : consider the union of the elements $x\in A$ : $\cup_{x\in A}B(x,\epsilon_x)$.
My problem is the following : the fact that $A$ is open gives us, for each $x\in A$. the existence of an epsilon such that….but there is no uniqueness. So when we take the union over the $x\in A$ how is chosen the $\epsilon_x$ if there are several ?
Thank you a lot