Is the Notion of the Empty Set Relative or Absolute? Suppose we specify subsets of a reference set by pairs, where the first co-ordinate specifies a member of the universe of discourse, and the second co-ordinate specifies the value that the characteristic function yields for that member of the reference set.  
E. G., if we have $\{a, b\}$ as the universe of discourse, then we have subsets $\{(a, 0), (b, 0)\}, \{(a, 0), (b, 1)\}, \{(a, 1), (b, 0)\}, \{(a, 1), (b, 1)\}$, where $(x, 0)$ indicates that $x$ does not belong to the set, while $(x, 1)$ indicates that $x$ belongs to the set.
Now, consider the empty set compared across different universes of discourses, for example $\{a, b\}$, and $\{a\}$.
For $\{a, b\}$ we have $\{(a, 0), (b, 0)\}$ as the empty set under this specification, and for $\{a\}$ we have $\{(a, 0)\}$ as the empty set under this specification.  
So, it would seem that we have a relative notion of an empty set in this context, but oftentimes books talk of "the empty set" which suggests it as absolute.  
So, does the concept of the empty set come as absolute, or relative?
 A: If you require extensionality from $\in$, i.e. two sets are equivalent if and only if they have the same memebers, and a model is transitive (a member of a set in the model is also in the model) - then the empty set is absolute.
This is quite simple to prove, since all the elements of all the sets are also sets, and the empty set is such that no one is a member of it.
On the other hand, if you consider $V$ a model of $ZF$, pick $x\in V$, and declare $\in^*$ to be the relation defined only on sets generated from iterations of $\mathcal P(x)$ (i.e. repeating the power set operation), no one will be in $x$, you will have a model of $ZF$ and the empty set of this model will be $x$.
However, as the intuition guiding us with topological spaces is mostly $\mathbb R^n$ the same happens with set theory, and the intuition that guides us is that of well founded and transitive models of $ZF$, in which the empty set is somewhat absolute (with respect to inner models and forcing extensions).
So essentially this all boils to your theory of sets, and how you define $\in$, and which models you are taking.
Addendum: An important point is internal and external view of the empty set. If a model is extensional (i.e. $\in$ satisfies extensionality) then the empty set is unique for the model, and all the models perceive their empty sets the same way -- the only set that has no elements (and this is unique by extensionality).
However, if we consider a model from an external point of view we have that it could have an empty set different than the model we work in, as the example I gave above. 
A: There are various ways to answer this question depending on what kind of perspectives you want to take. First, I wouldn't call what you're describing an empty set, exactly: since you're specifying the superset, you're really describing the unique inclusion map $\emptyset \to A$, which has $A$ as part of its data. 
Taking a foundational point of view, it is true that in ZF there is literally a unique empty set $\{ \}$ (it exists by the empty set axiom and is unique by the axiom of extension). The empty subset of any set is precisely this empty set, although in ZF subsets do not come with an identification of their parent set, and if you provide such an identification you are specifying a function of some kind like I said above. 
Taking a more categorical view, "the" empty set is "the" initial object in the category of sets. Initial objects are not literally unique in general, but an initial object is more than an object: it comes with distinguished maps $\emptyset \to A$ for every set $A$ satisfying the appropriate universal property, and any two initial objects (together with those maps) are isomorphic via a unique isomorphism. This is one way to rigorously justify the use of "the" empty set even if you aren't taking a foundational point of view. 
From the categorical point of view, one can think of subsets of a set $A$ as monomorphisms $S \to A$, and then associated to any set $A$ is the unique morphism $\emptyset \to A$, which one can think of as the initial object in the category of monomorphisms into $A$ (or more generally the category of morphisms into $A$). So in some sense it is the "relative empty set" over $A$. 
A: It is absolute. What you're seeing is an image of the empty set under a map and this can be different for different maps.
