Definition:Separation (Topology)

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

A partition $A \mid B$ of $T$ is a pair of open sets $A, B \in \tau$ such that:
 * $A$ and $B$ are non-empty
 * $A \cup B = S$
 * $A \cap B = \varnothing$

That is, such that $A$ and $B$ form a (set) partition of the set $S$.

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

Also known as
A partition in this particular context is also known as a separation.

Also see
Having no partition is equivalent to being connected, a fact which is demonstrated in Equivalence of Connectedness Definitions.