My question is as the title, i.e., 


Is there a metaLindelof nonLindelof space which has a dense hereditarily Lindelof subspace?


The question is related to the following result:
Every separable metaLindelof space is Lindelof.


Thank you!



Is there a metaLindelof nonLindelof space which has a dense hereditarily Lindelof subspace?
Suppose $X$ is metaLindelöf, and $A \subseteq X$ is hereditarily Lindelöf and dense.


Let $\mathcal{U}$ be an open cover of $X$ and let $\mathcal{V}$ be a point-countable refinement of $\mathcal{U}$. By a standard fact on covers, there is a discrete subset $D \subseteq A$ such that $\operatorname{st}(D,\mathcal{V}\_A) = A$ (where $\mathcal{V}\_A$ is the relativised version of $\mathcal{V}$). As $A$ is hereditarily Lindelöf, $D$ is countable and as $A$ is dense, $\mathcal{V} = \bigcup\_{d \in D} \{V \in \mathcal{V}: d \in V\}$ and as $\mathcal{V}$ is point-countable, $\mathcal{V}$ is countable, and so $\mathcal{U}$ has a countable refinement which implies that $X$ is Lindelöf.


So there is no such example.


For similar ideas, see [here](https://math.stackexchange.com/a/595853/4280), or one of some recent papers on star-properties like [this survey](http://at.yorku.ca/v/a/a/a/19.htm),or e.g. [this paper](https://staff.science.uva.nl/j.vanmill/papers/papers2011/al-ju-vm-tk-wi.pdf) etc. 


