What is the general outline of the proof that the real line is connected?

Question

I am reading many topology books and I want to understand the proofs that the real line is connected.

What is the general structure and outline showing that the real line is connected?

Answer 1

Suppose towards contradiction that $\mathbb{R}=U\sqcup V$, $U$ and $V$ nonempty and open.

Then there are real numbers $x\in U$ and $y\in V$ - without loss of generality let's assume $x<y$.

Now - looking at the interval $[x, y]$ - let $$z=\inf\{a\in [x, y]: a\in V\}.$$ Such a real $z$ exists, by the completeness of $\mathbb{R}$.

Now, which piece of $\mathbb{R}$ is $z$ in?

• Well, by definition $z$ is a limit point of $V$ (from the right). But since $U$ is open, $V$ is closed, so $z$ must be in $V$!

• At the same time, since $z\in V$ we must have $x<z$; so $z$ is a limit point of $U$ (from the left). So since $V$ is open, $U$ is closed, so $z$ must be in $U$!

• But $U$ and $V$ are disjoint.