Is there a name for dividing a set into pieces, some of which may be empty?

Suppose that $X$ is a set and $V_{0}$, $J$, and $V_{1}$ are pairwise disjoint subsets of $X$ whose union is $X$. If the three subsets were nonempty it would be a partition of $X$. However, I wish to include the possibility that at least one of the sets, perhaps two of them, are empty.

For further context I work with abstract convexity. All three sets are convex and the union of $J$ with either $V_{i}$ is also convex. This is an abstraction of the following situation:

Let $f \colon V \rightarrow \mathbb{R}$ be a linear map from a real vector space to $\mathbb{R}$ and $c \in \mathbb{R}$. Define the three sets as follows:

$V_{0} = \{ v \in V \colon f(v) < c \}$

$J = \{ v \in V \colon f(v) = c \}$

$V_{1} = \{ v \in V \colon f(v) > c \}$

Is there a commonly used name for his situation?

• Something about equivalence classes induced by equality under $\operatorname{sgn}\circ f$?
– user856
Feb 5, 2016 at 16:19
• The system of abstract convexity I am interested in, usually it is called a closure system, has as axioms: The sets $\varnothing$ and $X$ are convex. If $\mathcal{C}$ is a collection of convex sets then so is $\cap \mathcal{C}$. The vector space example is an instance of the situation I am interested in.
– Jay
Feb 5, 2016 at 20:56