For instance, consider the space $C[0,1]$, I hope $f_n \rightarrow f$ means that $\sup_{[0,1]} |f_n - f| \rightarrow 0$.
I know the convergence is meaningful only by specifying the topology, and I am wondering whether I induce one topology by the specified convergence.
First, you don't really need the notion of topology to make sense of the notion of convergence. In fact, there are notions of convergence that are provably not-topological (like convergence almost everywhere). Non-topological notions of convergence arise naturally, so one may be lead to consider new notions of space, like convergence spaces, where convergence is primitive (as opposed to being induced by something else, like a topology). You might wanna take a look at the article An initiation into convergence theory. In the article the author also describes how a convergence space induces a topology on the underlying set, the problem is that then this topology will introduce a (possibly strictly) weaker notion of convergence than the one you started with (for example, read this question).