TITLE: Topological Characterisation of the real line.
QUESTION [39 upvotes]: What is a purely topological characterisation of the real line( standard topology)?

REPLY [8 votes]: Take the category of completely regular spaces (subspaces of compact Hausdorff spaces or uniformisable spaces, you do not need the real numbers) with continuous maps. The cogenerators in this category are precisely all spaces containing the real numbers as a subspace. Now you can choose all connected cogenerators which are contained in every cogenerator. You get (up to homeomorphisms) $[0,1]$, $[0,1)$ and $\mathbb{R}$. Only one of these objects can be embedded densely into the other two objects: $\mathbb{R}$. Notice that you can state this definition without refering to anything else than the morphisms in the category of completely regular spaces. This definition is quite amusing in my opinion, although it might not be very practical.