Most interesting topological spaces are Hausdorff. Hausdorffness is identified as being the second separation condition. The first condition, T1 is a sub-requirement of T2 (Housdorff), so it is useful to keep it in mind when thinking about the Hausdorff condition.
A topological space \( X \) is said to be \( T_1 \) iff [...].
Now for Hausdorff.
A topological space \( X \) is Hausdorff (or \( T_2 \)) iff [...].
Lemma. Every Hausdorff space is \( T_1 \).