Non-Hausdorff manifold

In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff.

Line with two origins
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, $$\R \times \{a\}$$ and $$\R \times \{b\}$$ (with $$a \neq b$$), obtained by identifying points $$(x,a)$$ and $$(x,b)$$ whenever $$x \neq 0.$$

An equivalent description of the space is to take the real line $$\R$$ and replace the origin $$0$$ with two origins $$0_a$$ and $$0_b.$$ The subspace $$\R\setminus\{0\}$$ retains its usual Euclidean topology. And a local base of open neighborhoods at each origin $$0_i$$ is formed by the sets $$(U\setminus\{0\})\cup\{0_i\}$$ with $$U$$ an open neighborhood of $$0$$ in $$\R.$$

For each origin $$0_i$$ the subspace obtained from $$\R$$ by replacing $$0$$ with $$0_i$$ is an open neighborhood of $$0_i$$ homeomorphic to $$\R.$$ Since every point has a neighborhood homeomorphic to the Euclidean line, the space is locally Euclidean. In particular, it is locally Hausdorff, in the sense that each point has a Hausdorff neighborhood. But the space is not Hausdorff, as every neighborhood of $$0_a$$ intersects every neighbourhood of $$0_b.$$ It is however a T1 space.

The space is second countable.

The space exhibits several phenomena that do not happen in Hausdorff spaces:


 * The space is path connected but not arc connected. In particular, to get a path from one origin to the other one can first move left from $$0_a$$ to $$-1$$ within the line through the first origin, and then move back to the right from $$-1$$ to $$0_b$$ within the line through the second origin.  But it is impossible to join the two origins with an arc, which is an injective path; intuitively, if one moves first to the left, one has to eventually backtrack and move back to the right.


 * The intersection of two compact sets need not be compact. For example, the sets $$[-1,0)\cup\{0_a\}$$ and $$[-1,0)\cup\{0_b\}$$ are compact, but their intersection $$[-1,0)$$ is not.


 * The space is locally compact in the sense that every point has a local base of compact neighborhoods. But the line through one origin does not contain a closed neighborhood of that origin, as any neighborhood of one origin contains the other origin in its closure.  So the space is not a regular space, and even though every point has at least one closed compact neighborhood, the origin points do not admit a local base of closed compact neighborhoods.

The space does not have the homotopy type of a CW-complex, or of any Hausdorff space.

Line with many origins
The line with many origins is similar to the line with two origins, but with an arbitrary number of origins. It is constructed by taking an arbitrary set $$S$$ with the discrete topology and taking the quotient space of $$\R\times S$$ that identifies points $$(x,\alpha)$$ and $$(x,\beta)$$ whenever $$x\ne 0.$$ Equivalently, it can be obtained from $$\R$$ by replacing the origin $$0$$ with many origins $$0_\alpha,$$ one for each $$\alpha\in S.$$  The neighborhoods of each origin are described as in the two origin case.

If there are infinitely many origins, the space illustrates that the closure of a compact set need not be compact in general. For example, the closure of the compact set $$A=[-1,0)\cup\{0_\alpha\}\cup(0,1]$$ is the set $$A\cup\{0_\beta:\beta\in S\}$$ obtained by adding all the origins to $$A$$, and that closure is not compact. From being locally Euclidean, such a space is locally compact in the sense that every point has a local base of compact neighborhoods. But the origin points do not have any closed compact neighborhood.

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

This is the quotient space of two copies of the real line $$\R \times \{a\} \quad \text{ and } \quad \R \times \{b\}$$ with the equivalence relation $$(x, a) \sim (x, b) \quad \text{ if } \; x < 0.$$

This space has a single point for each negative real number $$r$$ and two points $$x_a, x_b$$ for every non-negative number: it has a "fork" at zero.

Etale space
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.)

Properties
Because non-Hausdorff manifolds are locally homeomorphic to Euclidean space, they are locally metrizable (but not metrizable in general) and locally Hausdorff (but not Hausdorff in general).