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.
