Is there any condition for which a lindeloff space is a separable space?

Question

I try to find a property of a Lindelof space for which the space is separable.
We know a space that is Lindelof but not separable.
So I try to give a special property of that space for which the space is separable.

Answer 1

Usually conditions assuring a Lindelöf space is separable are trivial.

For instance, all spaces with countable network are both separable and Lindelöf (in fact, such regular spaces are exactly continuous images of separable metric spaces [Gru, 4.9]). For sufficiently good spaces we cannot obtain more because of the following generalization of the well-known for metrizable spaces fact.

Theorem. [Gru, 4.4] Let $X$ be a paracompact $\sigma$-space (in particular, a regular stratifiable space (by [Gru, 5.7 and 5.9]). Then the following are equivalent.

(i) $X$ is (hereditarily) Lindelöf.

(ii) $X$ is (hereditarily) separable.

(iii) $X$ is (hereditarily) ccc.

(iv) $X$ has a countable network.

Proof. We recall that a regular space $X$ is a $\sigma$-space if $X$ has a $\sigma$-discrete (equivalently, $\sigma$-locally finite network); a space $X$ satisfies ccc (Closed Countable Chain) condition if each family of its disjoint non-empty open subsets is (at most) countable. Any closed discrete collection of subsets of a Lindelöf, separable, or ccc collectionwise-normal space is easily seen to be countable. Thus (i), (ii), and (iii) each imply (iv). On the other hand, any space satisfying (iv) is easily seen to be Lindelöf, separable, and ccc. Since property (iv) is hereditary, the result follows. $\square$

On the other hand, even such good spaces as compact may be non-separable. I don't know where is situated the border of the domain of spaces which are separable, if Lindelöf.

References

[Gru] Gary Gruenhage Generalized Metric Spaces, p. 423-501 in: K.Kunen, J.E.Vaughan (eds.) Handbook of Set-theoretic Topology, Elsevier Science Publishers B.V., 1984.