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.

In constructive mathematics

In constructive mathematics, the real numbers no longer automatically satisfy analytic LPO, so the tight apartness relation $x \# y = x \gt y \vee x \lt y$ no longer coincides with denial inequality $\neg (x = y)$. One typically uses the tight apartness relation as the inequality $x \neq y$ of the real numbers.

Furthermore, in the absence of analytic LPO, the relation used to define the line with two origins is only a partial equivalence relation: it is only defined on the subset of real numbers where $x \# 0$ and $x = 0$. The partial equivalence relation is only an equivalence relation if and only if analytic LPO holds of the real numbers, making the subset of real numbers where $x \# 0$ and $x = 0$ an improper subset of the real numbers. Thus, one can only take the quotient set of this relation to form the line with two origins if and only if analytic LPO holds for the real numbers.

Related concepts

• long line

References

• Wikipedia, Line with two origins

See also:

• Igor Vlasenko, Sergiy Maksymenko: One-dimensional non-Hausdorff manifolds and CW complexes [arXiv:2604.21868]