Definition:Set Partition

Set Theory
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: $$\forall T_1, T_2 \in \mathbb{S}: T_1 \cap T_2 = \varnothing$$;
 * 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.

Topology
The definition of a partition in the field of topology is slightly more specialized, as follows.

Let $$T = \left\{{S, \vartheta}\right\}$$ be a topological space.

Then a partition $$A | B$$ of $$T$$ is a pair of subspaces $$A, B \subseteq T$$ such that:
 * $$A$$ and $$B$$ form a partition of the set $$S$$ (as in the above context);
 * Both $$A$$ and $$B$$ are open in $$T$$.

It follows that not only are $$A$$ and $$B$$ are open in $$T$$, they are also both (by definition) closed in $$T$$.