How to show $X$ is Lindelöf in its compactification? Let $Z$ be any Hausdorff compactification of $X$, each point $z$ of $Z\setminus X$ has a closed $G_\delta$-set $F$ such that $z \in F$ and $F\cap X=\emptyset$. How to show $X$ is Lindelöf? 
And if $X$ is Lindelöf, then for each point $z$ of $Z\setminus X$, does it have a closed $G_\delta$-set $F$ such that $z \in F$ and $F\cap X=\emptyset$? 
Thanks very much.
 A: $\newcommand{\cl}{\operatorname{cl}}$The second question is easier. Suppose that $X$ is Lindelöf, and fix $p\in Z\setminus X$. For each $x\in X$ let $U_x$ be an open nbhd of $p$ in $Z$ such that $x\notin\cl_ZU_x$, and let $V_x=X\setminus\cl_ZU_x$. Then $\{V_x:x\in X\}$ is an open cover of $X$, so there is a countable $C\subseteq X$ such that $\mathscr{V}=\{V_x:x\in C\}$ covers $X$. 
Enumerate $\mathscr{V}$ as $\{V_n:n\in\omega\}$, and let $W_0=V_0$. Given $W_n$ for some $n\in\omega$, let $W_{n+1}$ be any open nbhd of $p$ such that $\cl_ZW_{n+1}\subseteq W_n\cap V_{n+1}$. Let $G=\bigcap_{n\in\omega}\cl_ZW_n=\bigcap_{n\in\omega}W_n$; clearly $G$ is a closed $G_\delta$, and $p\in G\subseteq Z\setminus X$.
I’m still thinking about the first question.
A: Regarding your first question:
Suppose $Z=\beta X$ and replace the condition "each point $z$ of $Z∖X$ has a closed $G_δ$-set $F$ such that $z\in F$ and $F∩X=∅$" with "for each point $z\in Z\setminus X$ there is a zero set $F\subset \beta X$ such that $z\in F$ and $F∩X=∅$". 
Then $X$ is realcompact, which is equivalent to  "X is homeomorphic to a closed subspace of $\mathbb R ^n$".  So in particular $X$ would be Lindelöf.
This is weaker than what you wanted, but it may lead in the right direction.
