On compactifications of Lindelöf spaces

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).