@Gro-Tsen: Good point, and thanks for the correction. I edited accordingly.

Do you have a reference for that result of Woodin? Sounds very interesting.

@HanulJeon: If I'm understanding the question right, I think $\mathcal U$ is meant to be an ultrafilter on $\omega$, not on some forcing poset. For example, the Mathias or Silver forcings each have a factorization into two posets, of the form (sigma-closed)*(ccc), where the sigma-closed poset adds a Ramsey ultrafilter on $\omega$ and then the ccc poset adds a real. If $\mathcal U$ is this ultrafilter (and we're forcing over $M$), then $M[\mathcal U]$ contains no new reals, even though $M[G]$ does, because $M[\mathcal U]$ only sees the sigma-centered part of the forcing.

Laver certainly understood this fact about finite support iterations when he wrote his paper on the Borel conjecture (1976). I just checked, though, and I can't see any hint in his paper of who first proved it.

Hi Steven, did you mean to say "$< \kappa$" in your definition of $\kappa$-Lindelöf?

@Wojowu: That's cheating. :) Can you do it without cheating?

Do you know of any compact Hausdorff spaces without this property? I can't think of one.

@MikhailKatz: Thanks Mikhail. I was wondering what the "false ones" might mean. (As I mentioned in my comment, I do not speak French.)

For anyone else on here as ignorant of French as I am, here is the translation that google gave me: "Moreover, both the true roots and the false ones are not always real, but sometimes only imaginary."