nLab non-Hausdorff topological space

Idea

A non-Hausdorff topological space is a topological space which is not a Hausdorff topological space.

Examples

Since the full subcategory of Hausdorff topological spaces inside the category of topological spaces is reflective, all limits in Top of diagrams of Hausdorff spaces are again Hausdorff spaces, but for colimits this fails in general (“non-Hausdorff quotients”, the colimit in the subcategory is given by applying Hausdorff reflection to the colimit formed in Top).

Hence examples of non-Hausdorff spaces generically arise from forming quotient topological spaces of Hausdorff spaces:

Example

Consider the real line $\mathbb{R}$ regarded as the 1-dimensional Euclidean space with its metric topology and consider the equivalence relation $\sim$ on $\mathbb{R}$ which identifies two real numbers if they differ by a rational number:

$$\left( x \sim y \right) \;\Leftrightarrow\; \left( \underset{p/q \in \mathbb{Q} \subset \mathbb{R}}{\exists} x = y + p/q \right) \,.$$

Then the quotient topological space

$$\mathbb{R}/\mathbb{Q} \;\coloneqq\; \mathbb{R}/\sim$$

is a codiscrete topological space, hence in particular a not a Hausdorff space.

Example

The line with two origins.