Classification of connected subsets of the real line (up to homeomorphism) I want to describe, up to homeomorphism, all the proper connected subsets of $\mathbb{R}$. I know the theorem that $A \subset \mathbb{R}$ is connected if and only if $A$ is an interval. So consider the following intervals: $[a,b), [a,b], (a,b)$ Note $[a,b)$ is not homeomorphic to $[a,b]$ (compactness argument) and $[a,b)$ is not homeomorphic to $(a,b)$ (by connectedness). Similary $[a,b]$ is not homeomorphic to $[a,b)$ (remove a,b from [a,b]). Now any ray $(a,\infty)$ is homeomorphic to $(1,\infty)$ which is homeomorphic to $(0,1)$ and $(0,1)$ is homeomorphic to $(a,b)$. Similarly the ray $[-a,\infty)$ is homeomorphic to $(0,1]$. So I think this covers all the cases. So I believe the answer is $3$, yes?

Not quite. It’s actually five:

1. the type of $(0,1)$ (and all open intervals and rays, including $\mathbb{R}$ itself);
2. the type of $[0,1)$ (and all half-open intervals and closed rays);
3. the type of $[0,1]$ (and all non-trivial closed intervals);
4. the type of $\\{0\\}$ (and all singletons); and
5. the type whose only representative is the empty set.



Those like Mariano who exclude $\varnothing$ from the class of connected sets will have only the first four types.