Non-Hausdorff manifold

The most familiar non-Hausdorff manifold is the line with two origins, or bug-eyed line.

This is the quotient space of two copies of the real line

$${\displaystyle \mathbb {R} \times \{a\}\quad {\text{ and }}\quad \mathbb {R} \times \{b\}}$$

with the equivalence relation

$${\displaystyle (x,a)\sim (x,b)\quad {\text{ if }}\;x\neq 0.}$$

This space has a single point for each nonzero real number ${\displaystyle r}$ and two points ${\displaystyle 0_{a}}$ and ${\displaystyle 0_{b}.}$ A local base of open neighborhoods of ${\displaystyle 0_{a}}$ in this space can be thought to consist of sets of the form

$${\displaystyle \{r\in \mathbb {R} \setminus \{0\}~:~-\varepsilon <r<\varepsilon \}\cup \{0_{a}\},}$$

where ${\displaystyle \varepsilon >0}$ is any positive real number. A similar description of a local base of open neighborhoods of ${\displaystyle 0_{b}}$ is possible. Thus, in this space all neighbourhoods of ${\displaystyle 0_{a}}$ intersect all neighbourhoods of ${\displaystyle 0_{b},}$ so the space is non-Hausdorff. The space is however locally Hausdorff in the sense that each point has a Hausdorff neighbourhood.

Further, the line with two origins does not have the homotopy type of a CW-complex, or of any Hausdorff space.[1]

Similar to the line with two origins is the branching line.

This is the quotient space of two copies of the real line

$${\displaystyle \mathbb {R} \times \{a\}\quad {\text{ and }}\quad \mathbb {R} \times \{b\}}$$

with the equivalence relation

$${\displaystyle (x,a)\sim (x,b)\quad {\text{ if }}\;x<0.}$$

This space has a single point for each negative real number ${\displaystyle r}$ and two points ${\displaystyle x_{a},x_{b}}$ for every non-negative number: it has a "fork" at zero.

The etale space of a sheaf, such as the sheaf of continuous real functions over a manifold, is a manifold that is often non-Hausdorff. (The etale space is Hausdorff if it is a sheaf of functions with some sort of analytic continuation property.)[2]