Embedding Regular Lindelöf Space in a Hausdorff Space

Question

Preliminary definition -

$G_{\delta}$ Closed: A set $A$ is $G_{\delta}$ closed if each point $x\not\in A$ is contained in a $G_{\delta}$ set disjoint from $A$.

Willard states that -

A regular space is Lindelöf iff whenever it is embedded in a Hausdorff space, it is $G_{\delta}$ closed.

With the hints given in the text, I've been able to prove the '$\Leftarrow$' implication. However, I'm having trouble with the '$\implies$' implication. Any help would be greatly appreciated!

Answer 1

Let $A$ be a regular Lindelöf space embedded in the Hausdorff space $X$. Let $x\in X\setminus A$, we want to construct a $G_\delta$ around $x$ disjoint from $A$.

For each $a\in A$ let $U_a$,$V_a$ be disjoint open sets containing $a$ and $x$ respectively. Since $A$ is Lindelöf there is a countable family $\{a_i\mid i\in\Bbb N\}$ such that $U_{a_i}$ covers $A$. The desired $G_\delta$ set around $x$ is then $\bigcap V_{a_i}$.