On compactifications of Lindelöf spaces

Question

Consider a compact Hausdorff space $X$ and let $A$ be a $G_\delta$ subset of $X$. Then the space $X\setminus A$ must be Lindelöf.

Keeping this in mind, I wonder if the converse is true, that is, if any Lindelöf Hausdorff space X admits a compactification $Y$ such that the remainder $Y\setminus X$ is a $G_\delta$ subset of $Y$. I'm familiar with the Stone-Cech compactification, but I can't see whether the remainder $\beta X\setminus X$ is a $G_\delta$ subset of $\beta X$.

Answer 1

Let $Y$ be any compactification of $X$. If $Y\setminus X$ is a $G_\delta$ subset of $Y$ then $X$ is an $F_\sigma$ subset of $Y$. Since $Y$ is compact this means $X$ is $\sigma$-compact. So for a counterexample let $X$ be a separable metric space which is not $\sigma$-compact, say $X=\mathbb R^\omega$ (Hilbert space) or $X=\mathbb N^\omega$ (the irrational numbers).

Answer 2

No, there are Lindelöf spaces that are not Čech-complete. E.g. The Sorgenfrey line is such an example.