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.
Assume there is a path from a = (1,0) to b = (omega_1,0).
Since the long line is Hausdorff, there is an arc from a to b.
Show the arc is increasing and use that to construct an order
embedding of omega_1 + 1 into [0,1], an impossibility.