Given a set or space X, a characteristic function on X is a function whose domain is X and whose value is either 0 or 1. The subsets of X may be taken as defined by characteristic functions on X.
It is usually assumed that characteristic functions are total, that is, defined for each member x of X. But there are also partial functions with domain of X and range of {0,1}. These may be regarded as defining subsets of X, which might be called "partial subsets."
There are hints of this sort of thing in, for example, Chapter 7 of Shoenfield's text Mathematical Logic. I'm not aware of any in-depth explorations of partial subsets, though, and would appreciate hearing about any you know of.