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.