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It is easy to see that every connected $1$-dimensional second-countable manifold (that is, what is often called just a manifold) is either homeomorphic to $\mathbb{R}$ or to $S^1$. Now let's drop the secound-countable-condition.

Is

How do you prove that every connected $1$-dimensional manifold homeomorphic to $\mathbb{R}, S^1$, the long line , or the long ray, or ? And why are the long circleline and the long ray not homeomorphic?

A good survey about the latter spaces can be found in the wikipedia entry. Basically, a long ray is built up of $\omega_1$-many intervals pasted together, and the the long line consists of two long rays in both directions. The long circle is constructed from the long line exactly as the circle $S^1$ is constructed from the line $\mathbb{R}$.
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It is easy to see that every connected $1$-dimensional second-countable manifold (that is, what is often called just a manifold) is either homeomorphic to $\mathbb{R}$ or to $S^1$. Now let's drop the secound-countable-condition.

Is every connected $1$-dimensional manifold homeomorphic to $\mathbb{R}, S^1$, the long line, the long ray, or the long circle?

A good survey about the latter spaces can be found in the wikipedia entry. Basically, a long ray is built up of $\omega_1$-many intervalls intervals pasted together, and the the long line consists of two long rays in both directions. The long circle is constructed from the long line exactly as the circle $S^1$ is constructed from the line $\mathbb{R}$.
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Classification of 1-dimensional manifolds (not second-countable)

It is easy to see that every connected $1$-dimensional second-countable manifold (that is, what is often called just a manifold) is either homeomorphic to $\mathbb{R}$ or to $S^1$. Now let's drop the secound-countable-condition.

Is every connected $1$-dimensional manifold homeomorphic to $\mathbb{R}, S^1$, the long line, the long ray, or the long circle?

A good survey about the latter spaces can be found in the wikipedia entry. Basically, a long ray is built up of $\omega_1$-many intervalls pasted together, and the the long line consists of two long rays in both directions. The long circle is constructed from the long line exactly as the circle $S^1$ is constructed from the line $\mathbb{R}$.