Line with two origins

The line with two origins is the real line $ℝ$ with two copies of the origin. #m/def/topology Consider the coproduct topology on $ℝ ⨿ ℝ$ with inclusions $𝜄 1, 𝜄 2 : ℝ ↪ ℝ ⨿ ℝ$ , and let $𝜄 1 (𝑥) \sim 𝜄 2 (𝑥)$ iff $𝑥 \neq 0$ . The line with two origins is then

𝐿 = (ℝ ⨿ ℝ) / \sim

We denote $𝜄 1 (0)$ by $01$ and $𝜄 2 (0)$ by $02$ . For all other $𝑥 \in ℝ$ we have $𝜄 1 (𝑥) \sim 𝜄 2 (𝑥)$ so we may unambiguously refer to $𝑥 \in ℝ$ .

Properties

$𝐿$ is path-connected but not arc-connected, for any path from $01$ to $02$ crosses itself.
$𝐿$ is a Non-Hausdorff manifold.

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