$\omega_1$ is $C^*$-embedded in $\omega_1+1$

Question

consider the space $ω_1$ (the first uncountable ordinal) and $ω_1+1$ together with the order topology. is it true that

1. $ω_1$ is $G_\delta$-dense subspace of $ω_1+1$?
2. $\omega_1$ is $C^*$-embedded in $\omega_1+1$?

If both proposition are correct, then we conclude that $\omega_1$ is $C$-embedded in $\omega_1+1$.

thanks.

Answer 1

Every continuous $f:\omega_1\to\Bbb R$ is eventually constant, so $\omega_1$ is indeed $C$-embedded in $\omega_1+1$; you’ll find a proof of the first statement here in Dan Ma’s Topology Blog.

1. Yes, this is true, since any $G_\delta$ in $\omega_1+1$ containing the point $\omega_1$ contains an interval $(\alpha,\omega_1]$ for some $\alpha<\omega_1$.

2. This is also true; in fact, $\omega_1+1=\beta\omega_1$.