6
$\begingroup$

This question fits the Generalised Baire space area. I am interested in the meagre ideal on ${}^\kappa \kappa$, with the bounded topology (or box topology), when, say, $\kappa$ is inaccessible.

To be more precise, basic open sets have the form $[s]=\{t\in {}^\kappa \kappa \mid t\supseteq s\}$, where $s\in {}^{<\kappa}\kappa$. A set $X$ is nowhere dense if every open set has an open subset that does not meet $X$. Finally, $X$ is meagre if it is the $\kappa$-union of meagre sets.

Are there examples or any criterion for a forcing $\mathbb{P}$ to satisfy the following property: "every meagre set in the generic extension is a subset of a meagre set in the sense of the ground model".

$\endgroup$
8
$\begingroup$

This is just a comment, not an answer, but the system does not allow me to comment:

(1) The proof of Claim 5.3(3) in Shelah's 1004 ( http://front.math.ucdavis.edu/1202.5799 ) may as usual give that "old meager sets are cofinal" is the same as "old non-meager sets are non-meager and old reals are dominating"

(2) The A-bounding forcings of Roslanowski/Shelah 860 ( http://front.math.ucdavis.edu/0508.5272 ) and more should fit the bill.

(3) Concerning "preserving non-meagerness" (not related here, but...) I would like to point out the iterable "manageable" condition from section 5 of Matet/Roslanowski/Shelah 799 (http://front.math.ucdavis.edu/0210.5087 )

Andrzej

$\endgroup$
  • $\begingroup$ This would be too long for a comment anyway... $\endgroup$ – Asaf Karagila Oct 21 '17 at 16:13

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.