What happens when there is uniform convergence on an unspecified set?

Question

Let $\{f_n\}$ define a sequence functions between metric spaces. I know what it means to say that

"$f_n$ converges uniformly on some set $U$.".

However, what if no set is specified, what does the sentence

"$f_n$ converges uniformly."

tell us about where $f_n$ converges uniformly? When no subset is specified; does that mean it converges uniformly on all of $X$? Or does it just mean that there exist some open set on which it converges uniformly?

Answer 1

Unless there is some context surrounding the question, allowing you to deduce $U$, I would take for $U$ the intersection of the domains of definition of $f_n$. Since usually they all have the same domain $X$ (which in this case seems to be one and the same metric space), I would say that your problem is about the convergence in the $\sup$ norm on $X$.