There are various equivalent definitions of a topological space and for some of them we have the concept of basis: a basis of opens sets, or a basis of neighbourhood.
This concept simplifies verifying continuity to a great degree.
So being in the most simple case possible: given a sequence space, what is a subset of sequences over which checking the continuity is sufficient?
For the general statement. Given a topological space $(X,\mathrm{Nets}_X)$ where $\mathrm{Nets}_X$ are all the converging nets of $X$, what is a subclass of $\mathrm{Nets}_X$ over which we can check continuity?