Partial subsets of a given set: reference request 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.
 A: In computability theory, the pervasive distinction between a computably decidable set and a computably enumerable set is exactly related to the concept you mention, namely, 


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*A set $A\subset\mathbb{N}$ is computably decidable if the characteristic function of $A$ is a computable function. 

*The set $A$ is merely computably enumerable, however,  if the partial function $1\upharpoonright A$, having constant value $1$ on domain $A$, is computable. Equivalently, $A$ is enumerable if there is some computable partial subset function, in your terminology, which includes $A$ in its domain and is correct for $A$ on its domain.  
The point is that not every computable partial function can be extended to a computable total function.
In set theory, one often considers the partial subset concept that you mention in connection with forcing. For example, the partial order of all finite partial subsets of $\mathbb{N}$, ordered by extension, is exactly the forcing to add a Cohen real. The idea is that partial information about the generic subset can be specified by a finite piece of it, and continued in diverse ways. Other forcing notions also use partial functions in this way, to allow a forcing condition to specify partial information about the generic object, often in combination with other complicated requirements on the domain or support of the conditions. 
