This is related to my [recent question](https://mathoverflow.net/questions/41597/transfinite-induction-a-theorem-of-pedersen-and-chains-of-subalgebras-of-bh) and would provide a natural positive answer to Question 2. I am sure this must be known to experts.



> 
> **Question:** Is there a monotone injection $(\omega\_1,<) \to ([0,1],<)$ ?
> 
> 
> 



Monotone injection of an ordinal into $[0,1]$
No, because you could use it to construct an injective map $\omega\_1\to\mathbb{Q}$, mapping $\alpha<\omega\_1$ to some rational number between $\alpha$ and $\alpha+1$.


