Let $X=\beta\omega\setminus\omega$ and let $Y=X\setminus P$ where $P$ is the set of P-points in $X$. Then $P$ is dense in $X$ if we assume CH but $P$ may be empty otherwise (Shelah). In the one case $Y$ is nowhere locally compact and in the other case $Y$ is a compact Hausdorff space. Elsewhere I have asked whether spaces like $Y$ (i.e. compact Hausdorff spaces with their P-points removed) are Baire spaces. Here I would like to ask about other properties of $Y$.
For example, $Y$ is a completely regular Hausdorff space, come what may. Is $Y$ always normal? Is $Y$ always Lindelof/weakly Lindelof? Is the ring of bounded continuous functions on $Y$ independent of the set theory in which one is working (equivalently, does $X$ always equal $\beta Y$)?