Suppose we have a Banach space $X$ and have chosen a set $\Sigma$ consisting of some sequences whose members are in $X$. We can then say that $(x\_n)\_{n=1}^\infty\in X^\mathbb{N}$ is $\Sigma$-*convergent* to $x\in X$ if $(x\_n-x)\_{n=1}^\infty\in \Sigma$. We can say $C\subset X$ is $\Sigma$-*closed* if whenever $(x\_n)\_{n=1}^\infty\subset C$ is $\Sigma$-convergent to $x\in X$, then $x\in C$. We can say $U\subset X$ is $\Sigma$-*open* if its complement is $\Sigma$-closed. 


Under some mild conditions on $\Sigma$, the collection $\tau$ of $\Sigma$-open sets is a topology on $X$ and $(x\_n)\_{n=1}^\infty$ is convergent to $x$ in the topology $\tau$ if and only if it is $\Sigma$-convergent to $x$. 


Is there some reference for this process? It seems to me that this must be a well known, standard procedure for generating a topology. 



Defining a topology by sequences
What you are describing is very close to what is called a convergence space. Strictly speaking, these are quite a lot more general than a topology, but with some additional conditions on the convergent filters, they form a topology and vice-versa. Several good references are "Foundations of Topology", by Gerhard Preuss, and "Convergence Structures and Applications to Functional Analysis", by Beattie and Butzmann.


