**EDIT:** I found a construction, see below. I decided not to delete the question in case someone is interested.

A space $X$ is *of Type I* if $X=\cup_{\alpha<\omega_1} X_\alpha$, where each $X_\alpha$ is open in $X$, $\overline{X_\alpha}\subset X_\beta$ whenever $\alpha<\beta$, and each $\overline{X_\alpha}$ is Lindelöf. Although it is not a part of the usual definition, I assume that Type I spaces are *not* Lindelöf, so $X\not=X_\alpha$ for any $\alpha$.
Many classical spaces of set-theoretic topology are of Type I: $\omega_1$ itself, the long line and the long ray, Aronszajn and Suslin trees with the order topology, etc. However these examples are all first countable, and I am trying to see whether there are Type I subspaces of $\omega^\ast=\beta\omega-\omega$ but have some trouble since I am not at all familiar with this space.

Is there (in ZFC) a Type I subspace of $\omega^\ast$ ?

** CONSTRUCTION:** *There is a countably compact Type I subspace X of $\beta\omega$ such that $X_\alpha$ is Lindelöf and $X_{\alpha+1}$ compact for each $\alpha$.*

Given any $p$ in $\omega^\ast$, define $\mathcal{U}=\{U\subset\omega^\ast\\,:\\,U$ is clopen and $p\not\in U\}$. Then $\mathcal{U}$ is a cover of $\omega^\ast -\{p\}$. It is shown in van Mill's article in the Handbook of set theoretic topology (Corollary 1.5.4) that no countable subfamily of $\mathcal{U}$ has a dense union in $\omega^\ast$. I will define $X_\alpha$ by induction, each will be an at most countable union of members of $\mathcal{U}$ and thus none will be dense in $\omega^\ast-\{p\}$.

Set $X_0 = \emptyset$. Given $X_\alpha$, take a finite family $\mathcal{F}\subset\mathcal{U}$ that covers the compact set $\overline{X_\alpha}$ and some $U\in\mathcal{U}$ not included in $\cup\mathcal{F}$. There is such a $U$ because $\cup\mathcal{F}$ is not dense and $\mathcal{U}$ is a cover of $\omega^\ast-\{p\}$. Set $X_{\alpha+1}=U\bigcup\cup\mathcal{F}$. Then $X_{\alpha+1}$ is clopen (and thus compact). For limit $\alpha$ define $X_\alpha=\cup_{\beta<\alpha}X_\beta$, then $X_\alpha$ is Lindelöf ($\sigma$-compact, actually) and not dense in $\omega^\ast-\{p\}$. Take $X=\cup_{\alpha<\omega_1}X_\alpha$, then $X$ is a Type I space which is countably compact since any countably infinite subset lies in some $X_{\alpha+1}$ which is compact, so there is an accumulation point.