# Extended Long Line isn't path-connected; proof

I know this is a well known result yet I'm having trouble proving it.

The open long line is defined as:

$L = \omega_1 \times [0,1)$ without its minimal element where $\omega_1$ is the first uncountable ordinal

The extended open long line is:

$L$*$= \omega_2 \times [0,1)$ without its minimal element, where $\omega_2$ is the successor of $\omega_1$.

Can you give me a hint as to how to prove $L$* isn't path connected, without using any notions of compactness? I know I need to choose points like $\{\{\emptyset\},0\}$ and $\{\omega_1,0\}$ and I thought of using the intermediate value theorem - but then a cardinality argument doesn't suffice.

• Be careful. Those two points are (1,0) and (omega_1,0). – William Elliot May 10 '17 at 10:05
• Not an answer but I think it is very interesting that the extended long line is "not path-connected", not because it has some gap or something, but just because it is (somehow) too long for any path to reach from one end to the other. This might be more of a flaw of the paths we are considering, than a flaw of $L^*$. – M. Winter May 10 '17 at 10:54