[[Topology counterexamples MOC]]
# Line with two origins

The **line with two origins** is the real line $\mathbb{R}$ with two copies of the origin. #m/def/topology 
Consider the [[coproduct topology]] on $\mathbb{R} \amalg \mathbb{R}$ with inclusions $\iota_{1},\iota_{2} : \mathbb{R} \hookrightarrow \mathbb{R} \amalg \mathbb{R}$,
and let $\iota_{1}(x) \sim \iota_{2}(x)$ iff $x \neq 0$.
The line with two origins is then
$$
\begin{align*}
L = (\mathbb{R} \amalg \mathbb{R}) / {\sim}
\end{align*}
$$
We denote $\iota_{1}(0)$ by $0_{1}$ and $\iota_{2}(0)$ by $0_{2}$.
For all other $x \in \mathbb{R}$ we have $\iota_{1}(x) \sim \iota_{2}(x)$ so we may unambiguously refer to $x \in \mathbb{R}$.

## Properties

- $L$ is [[Path connectedness|path-connected]] but not [[Path connectedness|arc-connected]], for any path from $0_{1}$ to $0_{2}$ crosses itself.
- $L$ is a [[Non-Hausdorff manifold]].


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