Almost all the spaces encountered in mathematical analysis are Hausdorff spaces; The most important real number is Hausdorff space. More generally, all metric spaces are Hausdorff spaces. In fact, many spaces used in analysis, such as topological groups and topological manifolds, explicitly declare Hausdorf conditions in their definitions.

The simplest example of topology in T 1 space instead of T2 space is cofinite space.

Pseudometric spaces are not typical Hausdorff spaces, but they are pre-regular, and are usually only used to construct Hausdorff gauge spaces in analysis. In fact, when analysts deal with non-Hausdorff space, it should at least be pre-regularized, and they just replace it with its Kolmogorov quotient space.

On the contrary, non-pre-regular spaces often appear in abstract algebra and algebraic geometry, especially as Zariski topology on algebraic clusters or commutative rings. They also appear in the model theory of intuitive logic: all complete Heyting algebras are open set algebras in a topological space, but this space does not need pre-regularization, let alone Hausdorff space.