In mathematics, specifically set theory, the term maximal set is often used to define specific sets in an elegant manner. In layman's terms, a maximal set describes a set that is "as large as possible". Without using this term, definitions of particular sets can get very cumbersome.
Specifically, suppose we want to describe a set A that satisfies some property P, but so that no proper subset of A should be included in the definition of such a set. Put another way, we want a set A satisfying the following two conditions:
- the set A has property P;
- if B is another set with property P, and if B contains A, then B must be equal to A.
Then we simply say that A is a maximal set that satisfies property P.
Note that maximal sets need not be unique. That is, there can be multiple maximal sets.
As an example, we can define a Black chain to be a maximal set of connected Black intersections. We consider each chain as a single unit, and we don't call a three-stone connected subset of a five-stone chain a "chain".
Note also that maximal does not mean the same as biggest (in the sense of largest number of elements). For example, suppose there is a five-stone chain on the board as described above, and there is also a six-stone chain elsewhere on the board. Then both chains are maximal in the sense described here. The six-stone chain has a larger number of stones than the five-stone chain, but it does not contain it.
 Mathematically speaking, "B contains A" means that every member of A is also a member of B; this includes the possibility that A and B are in fact equal.