An example about Lindelöf spaces

Question

Let $X$ a Lindelöf space and $A \subset X$ a closed subspace, we have seen here

Closed subsets of Lindelöf spaces are Lindelöf

that $A$ need to be Lindelöf. My question is about the reciprocal : if $A$ is a Lindelöf subspace of a Lindelöf space $X$, then $A$ need to be closed??

I guess not, but I cannot find an example, cause everithing I need is about compact sets.

Can you give me some example?

Answer 1

No. A space in which every subspace is Lindelöf is called hereditarily Lindelöf. For example, any separable metrizable space. Or any 2nd-countable space. Or any countable space.

Answer 2

Since a countable union of Lindelöf subspaces is Lindelöf, if $X$ is a Lindelöf space in which every Lindelöf subspace is closed, then $X$ has the property "every countable union of closed sets is closed," equivalently, "every countable intersection of open sets is open." Such spaces are called P-spaces (in the sense of Gillman and Henriksen), so $X$ must be a Lindelöf P-space. A converse holds for Hausdorff spaces: a Lindelöf subspace of a Hausdorff P-space is closed, by a similar argument to the proof that a compact subspace of a Hausdorff space is closed.

For a concrete counterexample to your question, take any Lindelöf space which is not a P-space, e.g., the space $\mathbb Q$ of rational numbers.