relationship between topology and convergences In my textbook, the topology of $C_c^\infty$ and $C^\infty$ are defined by: how $f_n\to f$ in these topologies. Similarly, the weak-* topology are also defined by the same manner.
I wonder why we can determine a topology only by convergent sequence since topology ought to be defined by open sets?
 A: What Owen said is correct.  It is not sufficient to specify a topology by defining what it means for sequences to converge.  For example, the discrete topology and the cocountable topology on $\mathbb{R}$ yield nonhomeomorphic spaces, but their notions of sequential convergence are the same:  a sequence converges iff it is eventually constant.
In practice, when defining topologies in this way, the definition of convergence you want to make for sequences still makes sense if you replace the word "sequence" with the word "net", and so fixing this is quite easy.  In particular, in your case, if you replace the word "sequence" with the word "net", there should be a unique topology on $C_c^{\infty}$ and $C^{\infty}$ whose notion of convergence agrees with your given condition.
Of course, you can't just make any old definition of convergence.  In order for it to define a topology, your definition has to satisfy a list of axioms---see, for example, Theorem 3.4.8 on pg. 105 here (note that if you're reading this in the future, these numbers may have changed).  This theorem gives the precise formulation of how one may define topologies by declaring what it means for nets to converge.
