It is fundamental to topology that $\mathbb{R}$ is a connected topological space. However, all the topology books that I have ever looked in give the same proof. (the proof I am thinking of can be seen in Munkres's topology or Lee's Introduction to topological manifolds)


This seems strange to me, because for other fundamental results such as the Compactness of $[0,1]$, I can think of several proofs. 


Does anyone know any different proofs of the connectedness of $\mathbb{R}$?



Connectedness and the real line
If you've already developed basic facts about compactness you can prove it this way:


Let $[0,1] = A \cup B$ with $A$ and $B$ closed and disjoint. Then since $A \times B$ is compact and the distance function is continuous, there is a pair $(a, b) \in A \times B$ at minimum distance. If that distance is zero, $A$ and $B$ intersect. If not, you get a contradiction by taking any point in the interval from $a$ to $b$: it can't be in either $A$ or $B$ because its distance from $b$ or $a$ is smaller than the minimum.


That shows a compact interval in $\mathbb{R}$ is connected. If $\mathbb{R} = A \cup B$ with $A$ and $B$ closed and disjoint, then for any closed interval $I$ with one endpoint in $A$ and one in $B$, $I = (A \cap I) \cup (B \cap I)$ is disconnection of $I$. Alternatively, you could write $\mathbb{R}$; as a union of closed intervals with a common point.


