Definition:Hausdorff Space

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

Also known as
This condition is known as the Hausdorff condition.

For short, $T$ is Hausdorff is used to mean $T$ is a Hausdorff space.

Conveniently, a topological space is Hausdorff if any two distinct points can be housed off from one another in separate disjoint open sets.

Some sources use the term separated space for Hausdorff space but this is discouraged as there already exists considerable confusion and ambiguity around the definition of the word separated in the context of topology.

Some authors require a space to be Hausdorff before allowing it to be classed as a topological space, but this approach is unnecessarily limiting.

Equivalence of Definitions
See Equivalence of Definitions of $T_2$ Space for a proof that these definitions are equivalent.

Note that while some sources give Definition 1 and others Definition 3, it is rarely indicated specifically that the definitions are equivalent. In particular, there may exist places on the internet where a page using Definition 3 may directly link to another page which uses Definition 1 without comment that the definitions are linguistically different.

Also see

 * Properties of Hausdorff Space