You can find an elementary proof in Corollary 4.4 of our paper joint paper:

• Mykola Lysynskyi, Sergiy Maksymenko, Classification of differentiable structures on the non-Hausdorff line with two origins, arxiv:2406.09576

It follows the line similar to the one mentioned in the answer by Will Sawin.

We needed that proof for having an equivalent statement which at first sight might look more unusual:

If $M$ is a one-dimensional (not necesarily Hausdorff) $C^{k}$-manifold and $U \subset M$ is an open subset homeomorphic with $\mathbb{R}$, then there exists a homeomorphism $h: U \to \mathbb{R}$, such that the chart $(h,U)$ is compatible with $C^k$-structure on $M$.

You can find an elementary proof in Corollary 4.4 of our joint paper:

• Mykola Lysynskyi, Sergiy Maksymenko, Classification of differentiable structures on the non-Hausdorff line with two origins, arxiv:2406.09576

It follows the line similar to the one mentioned in the answer by Will Sawin.

We needed that proof for having an equivalent statement which at first sight might look more unusual:

If $M$ is a one-dimensional (not necesarily Hausdorff) $C^{k}$-manifold and $U \subset M$ is an open subset homeomorphic with $\mathbb{R}$, then there exists a homeomorphism $h: U \to \mathbb{R}$, such that the chart $(h,U)$ is compatible with the $C^k$-structure on $M$.

You can find an elementary proof in our paper joint paper, Corollary 4.4:

• Mykola Lysynskyi, Sergiy Maksymenko, Classification of differentiable structures on the non-Hausdorff line with two origins, arxiv:2406.09576

It follows the line similar to the one mentioned in the answer by Will Sawin.

You can find an elementary proof in Corollary 4.4 of our paper joint paper:

• Mykola Lysynskyi, Sergiy Maksymenko, Classification of differentiable structures on the non-Hausdorff line with two origins, arxiv:2406.09576

It follows the line similar to the one mentioned in the answer by Will Sawin.

We needed that proof for having an equivalent statement which at first sight might look more unusual:

If $M$ is a one-dimensional (not necesarily Hausdorff) $C^{k}$-manifold and $U \subset M$ is an open subset homeomorphic with $\mathbb{R}$, then there exists a homeomorphism $h: U \to \mathbb{R}$, such that the chart $(h,U)$ is compatible with $C^k$-structure on $M$.