Is the union of less than $2^{\aleph_0}$ (closed) sets of empty interior, a set of empty interior?

Question

Baire category theorem gives this result for countable unions. But what can be said about unions of size less than $2^{\aleph_0}$?

I only need this to be true in $\mathbb{R}^2$, for my purposes.

Strictly speaking, I only need the following: An open set in $\mathbb{R}^2$ is not the union of less than $2^{\aleph_0}$ straight lines.

Answer 1

Meager sets, i.e. countable union of closed nowhere dense sets, have empty interior as you mentioned above.

The covering number of the meager ideal is the smallest cardinality $\kappa$ so that $\mathbb{R}$ is the union of $\kappa$ many meager set.

If the covering number of the meager ideal is less than $2^{\aleph_0}$, then $\mathbb{R}$ is the union of less than $2^{\aleph_0}$ meager sets. Replacing each meager set with the countably many closed nowhere dense set forming it will give you a less than $2^{\aleph_0}$ union of closed nowhere dense sets with empty interior.

This situation can occur but of course requires the failure of the continuum hypothesis.

Answer 2

Re: Strictly speaking, I only need the following: An open set in $\mathbb{R}^2$ is not the union of less than $2^{\aleph_0}$ straight lines.

This is true since for example: Every open set in plane contains a circle. And less than continuum lines cannot cover a circle.