Definition:Set Partition

Definition
Let $$S$$ be a set.

A partition of $$S$$ is a set of subsets $$\mathbb S$$ of $$S$$ such that:


 * 1) All sets in $$\mathbb S$$ are pairwise disjoint: $$\forall S_1, S_2 \in \mathbb S: S_1 \cap S_2 = \varnothing$$;
 * 2) The union of all the sets forms the whole set $$S$$: $$\bigcup \mathbb S = S$$;
 * 3) None of the sets in $$\mathbb S$$ is null: $$\forall T \in \mathbb S: T \ne \varnothing$$.

A partition is sometimes called a decomposition.

This same definition is sometimes also encountered in combinatorics.

Finite Expansion
If $$S_1, S_2, \ldots, S_n$$ form a partition of $$S$$, the notations:
 * $$S = S_1 | S_2 | \cdots | S_n$$

or:
 * $$S = \left\{{S_1 | S_2 | \cdots | S_n}\right\}$$

are sometimes seen.

The representation by such a partition $$\bigcup_{k=1}^n S_k = S$$ is also called a finite expansion of $$S$$.

Note
The definition of a partition in the field of topology is slightly more specialized.