Definition:Hausdorff Space/Equivalence of Definitions

Equivalence of Definitions of Hausdorff Space

See Equivalence of Definitions of $T_2$ Space for a proof that these definitions of a Hausdorff space 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.