4
$\begingroup$

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.

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

$\endgroup$

1 Answer 1

4
$\begingroup$

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).

$\endgroup$
2
  • 1
    $\begingroup$ 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 $n \in \mathbb{N}^+$. $\endgroup$ Jun 14, 2018 at 21:44
  • 1
    $\begingroup$ 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. $\endgroup$ Jun 14, 2018 at 21:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.