No covering with nowhere dense \textsf{P}-sets in the Cohen model

TL;DR

This paper demonstrates that in certain Cohen models, the space  can not be covered by nowhere dense P-sets if fewer than _{\u00a9} many Cohen reals are added, revealing limitations on covering properties in these models.

Contribution

It establishes a new result linking Cohen forcing with the non-coverability of  by nowhere dense P-sets under specific cardinal assumptions.

Findings

 cannot be covered by nowhere dense P-sets after adding fewer than _{a9} Cohen reals.

Existence of an ultrafilter on  without a tower in these models.

Connection between Cohen forcing and covering properties in topology.

Abstract

We prove that if less than $\aleph_{\omega}$-many Cohen reals are added to a model of \textsf{CH}, then $\omega^{\ast}$ can not be covered by nowhere dense \textsf{P}-sets (equivalently, there is an ultrafilter on $\omega$ that does not contain a tower).