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I am having some trouble with the following result presented here:

enter image description here

Obviously I'm missing something, but I think from that result it could be shown that if $X$ is an infinite topological space, then $d(X) \leq hL(X)$, where $d$ is the density and $hL$ is the hereditary Lindelöf degree. To see this simply observe that, by the above result, there exists a right-separated subspace $Y$ of $X$ with $d(X)\leq |Y|$. Then, since $hL(X) = \sup \{ |Z| : Z \ \text{is right-separated} \}+\omega$ (this is 2.9(b) in Juhász's book of cardinal functions), it follows immediately that $d(X)\leq hL(X)$. Of course, this is incorrect, any hereditarily Lindelöf non-separable space does not satisfy the previous inequality.

Naturally, I don't think Hajnal and Juhász are wrong, I just don't understand where I'm wrong.

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    $\begingroup$ Might this be a confusion between right and left separated? I have great difficulty remembering which is which, so I wouldn't be surprised to see different conventions in different papers or books. $\endgroup$
    – Andreas Blass
    Commented Jul 10, 2022 at 2:29
  • $\begingroup$ You are right, it is a matter of definitions. In the article I linked above they define "right-separated" as closed initial segments, but in Juhász's book (and several other articles by them both) "right-separated" is defined as open initial segments. I did not expect Juhász to use one definition in one place and another in another place. I should have read more carefully. Thank you very much for your reply! $\endgroup$
    – Peluso
    Commented Jul 10, 2022 at 18:05

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