What does partition BETWEEN two sets mean?

Question

Partition of a space is a just bunch of subsets of the space such as any element belongs to exactly one subset.

However, when we have two disjoint closed subsets A, B of some space X, than what is an arbitrary partition P between A, B? Does it mean that this partition randomly sorts the rest of X into either A, or B? When $A \cup B$ $\neq X$, should I understand it such as the partition P just creates two equivalence classes from all elements somehow?

Thank you very much.

Source: I have seen this in Van Mill´s book on topology of function spaces. Exact expression from the book:

Answer 1

I have found another definition of the same concept.

Let $X$ be a space. An closed set $L_i$ separates two closed subsets of $X$, $A_i$ and $B_i$ if $X - L_i$ can be written as a union of two separated sets, one containing $A_i$ and the other containing $B_i$. This is basically the claim "$L_i$ is partition between $A_i$ and $B_i$".

If the intersection of all such $L_i$ for all pairs $(A_i, B_i)$ is nonempty, then the family of pairs $(A_i, B_i)$ is called essential.

So the meaning of "partition between two sets" is a closed set such that when we "cut it out" of the space, the two sets can still be separated.

Source: LEONARD R. RUBIN - NONCOMPACT HEREDITARILY STRONGLY INFINITE DIMENSIONAL SPACES

Answer 2

This just means any choosing of $A_i$ and $B_i$. So it needs to hold for $$(\bigcap_{i \in G_1} A_i) \cap (\bigcap_{j \in G_2}B_i)$$ with $G_1 \dot{\cup} \ G_2 = G$