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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.

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

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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).

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    $\begingroup$ It's problem 18 on page 608 in Arhangel'skij's survey on $C_p$-theory: due to Velichko, let $C_p(X)$ be a hereditarily Lindelöf space. Is it true that $(C_p(X))^n$ is hereditarily Lindelöf for all $n \in \mathbb{N}^+$? By Zenor and Velichko's results (from 1980 resp. 1981) we know already that $(C_p(X))^n$ is hereditarily Lindelöf for all $n \in \mathbb{N}^+$ iff $X^n$ is hereditarily separable for all $n \in \mathbb{N}^+$. $\endgroup$
    – Henno Brandsma
    Commented Jun 14, 2018 at 21:44
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    $\begingroup$ The survey paper then goes on to discuss some results that are already known, e.g. if $C_p(X) \times C_p(X)$ is HL, then $X^n$ is HS for all $n$ etc. Also in any model without $S$-spaces the result is true, as Arhangel'skij has shown himself. $\endgroup$
    – Henno Brandsma
    Commented Jun 14, 2018 at 21:49

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