nLab line with two origins

Idea

What is called the line with two origins is the topological space which results by “gluing” a copy of the real line identically to itself, except at the origin. This is a basic example of a non-Hausdorff topological space (even a non-Hausdorff smooth manifold) and of the fact that quotient topological spaces of Hausdorff topological spaces need not be Hausdorff themselves.

Definition

Consider the disjoint union $\mathbb{R} \sqcup \mathbb{R}$ of two copies of the real line $\mathbb{R}$ regarded with its metric topology. Moreover, consider the equivalence relation on the underlying set which identifies every point $x_i$ in the $i$th copy of $\mathbb{R}$ ($i \in \{0,1\}$) with the corresponding point in the other, the $(1-i)$th copy, except when $x = 0$:

$$\left( x_i \sim y_j \right) \;\Leftrightarrow\; \Big( \left( x = y \right) \,\text{and}\, \big( ( x \neq 0 ) \,\text{or}\, ( i = j ) \big) \Big) \,.$$

The quotient topological space

$$\left( \mathbb{R} \sqcup \mathbb{R} \right)/\sim$$

by this equivalence relation is called the line with two origins.

Properties

The line with two origins is a non-Hausdorff topological space:

Because by definition of the quotient space topology, the open neighbourhoods of $0_i \in \left( \mathbb{R} \sqcup \mathbb{R} \right)/\sim$ are precisely those that contain subsets of the form

$$(-\epsilon, \epsilon)_i \;\coloneqq\; (-\epsilon,0) \cup \{0_i\} \cup (0,\epsilon) \,.$$

But this means that the “two origins” $0_0$ and $0_1$ may not be separated by neighbourhoods, since the intersection of $(-\epsilon, \epsilon)_0$ with $(-\epsilon, \epsilon)_1$ is always non-empty:

$$(-\epsilon, \epsilon)_0 \cap (-\epsilon, \epsilon)_1 \;=\; (-\epsilon, 0) \cup (0, \epsilon) \,.$$

The Hausdorff reflection

$$H \;\colon\; Top \longrightarrow Top_{Haus}$$

of the line with two origins is the real line itself:

$$H\left( \left( \mathbb{R} \sqcup \mathbb{R} \right)/\sim \right) \;\simeq\; \mathbb{R}$$

However, the line with two origins is T1 and sober.

It is also a topological manifold, even a smooth manifold, except that one often requires such to be Hausdorff to rule out examples such as this. In a similar vein, it is locally compact (and a compact version can be made by starting with $[{-1,1}]$ instead of $\mathbb{R}$), paracompact, and so forth.

Related concepts

• long line

References

• Wikipedia, Line with two origins