$[0,\omega_1]$ is not first countable.

Question

I was looking for examples of spaces that are not first countable. On wikipedia they give the example of $[0,\omega_1]$. I think I understand that $\omega_1$ is a limit point of $[0,\omega_1)$ since if $(\alpha,\beta)$ is an open containing $\omega_1$ it contains an ordinal smaller than $\omega_1$ and thus this ordinal is in $[0,\omega_1)$. Now why is there no sequence of $[0,\omega_1)$ converging to $\omega_1$. It is not entirely clear to me how to prove this. Also I don't really understand why this implies that $\omega_1$ doesn't have a countable basis.

Answer 1

If there exist a sequence $x_n\in [0,\omega_1)$ converging to $\omega_1$, then $\omega_1$ would be countable, which is not.

To see this you can use the model where ordinals are defined as

$0=\emptyset$

$1=\{\emptyset\}=\{0\}$

$2=\{\emptyset,\{\emptyset\}\}=\{0,1\}$

...

$\alpha=\{\beta: \beta<\alpha\}$ the set of ordinals smaller than $\alpha$

Then, if $x_n<\omega_1$ and $x_n\to \omega_1$ then $\omega_1=\cup_n x_n$ is coutable union of countable set (because $\omega_1$ is the first uncountable) and so it would be countable.