> - ∀x,y ∈ X with x ≠ y, ∃U,V ⊂ X s.t. x ∈ U, y ∈ V and U ∩ V = ∅. Amusingly and to go in the direction of your argument, this is not a very rigorous definitio

> - ∀x,y ∈ X with x ≠ y, ∃U,V ⊂ X s.t. x ∈ U, y ∈ V and U ∩ V = ∅. Amusingly and to go in the direction of your argument, this is not a very rigorous definition of a Hausdorff space. You should introduce the topology τ associated with X and specify that U and V are open sets of τ.