The passage is from "A Beginners Guide to Geometric Measure Theory" by Frank Morgan.
He defines a rectifiable current as:
"an oriented rectifiable set with integer multiplicities, finite area, and compact support. By general measure theory, one can integrate a smooth differential form $\phi$ over such an oriented rectifiable set S, and hence view S as a current, i.e. a linear functional on differential forms $$ \phi\mapsto \int_S \phi $$ This perspective yields a new natural topology on the space of surfaces dual to an appropriate topology on differential forms."
So, if I have understood correctly, what is being said is that picking some surface in n-space "is the same" or gives you a natural (meaning no choice must be made?) linear functional mapping differential forms to whatever scalar given by the above integral.
Assuming that is correct, is the salient point then that whatever topology you endow the surfaces in your space is then naturally passed down to differential forms?
I have also never heard of a topology on differential forms and googling doesn't yield much, what sort of book should I look in to learn about what a topology on forms does/looks like?
Sorry for the blocks of text and I appreciate any help.