Two set-theoretical assumptions

Question

My question about the following two set-theoretical assumptions:

1. union of less than continuum many meager subsets of $\Bbb R$ is meager in $\Bbb R.$

2. Union of less than continuum many meager subsets of $\Bbb R$ is not $\Bbb R.$

Clearly, (1) implies (2) but if there is another condition can be added to (2) so we can (2)+ condition implies (1).

Answer 1

Apart from trivially $(1)$ itself, there is $(3)$: "the union of less than continuum many compact subsets of $\omega^\omega$ is contained in a $\sigma$-compact set" that you can add to $(2)$ to imply $(1)$.

$(1)$ is known as $\operatorname{add}(\mathcal{M}) = \mathfrak{c}$, (2) is $\operatorname{cov}(\mathcal{M}) = \mathfrak{c}$ and (3) is $\mathfrak{b} = \mathfrak{c}$. A look at Cichoń's diagram shows that what I say is correct.

I don't know why are asking this, but it looks like it is set up to give this answer.