Let $X$ be a (infinite) separable topological space and consider $C\_p(X)$, the space of continuous functions on $X$ endowed with the point-wise convergence topology.



> 
> Q. I am looking for topological properties on $X$ which make $C\_p(X)$ hereditary Lindelöf.
> 
> 
> 


$$X=?\implies C\_p(X)=\textrm{Hereditary Lindelöf}$$ 



Hereditary Lindelöfness in $C_p$-spaces
If $X^n$ is hereditarily separable for each $n \in \mathbb{N}$ then $C\_p(X)$ is hereditarily Lindelof by Zenor-Velichko's theorem. It is consistent with ZFC that this is also a necessary condition and it was an open problem in the 80's to find a consistent counterexample. I don't know the status of this problem (I'm almost sure that it appeared in the first version of *Open problems in Topology*, but I can't check that right now).


