Let $M$ a countable transitive model $ZFC$.
In $M$, let $\mathbb{P}=Fn(\kappa,2)$, where $\kappa$ is any cardinal. Let $G$ be $\mathbb{P}$-generic over $M$. Then $M[G] \models \text{cov(meager)}\geq \kappa$.
Let $\mathcal{I}$ be a proper ideal of subsets of a set $X$, contanining all singletons of $X$.
The covering number $\mathcal{I}$, $\text{cov($\mathcal{L}$)}$, is the smallest number of the sets in $\mathcal{I}$ with union $X$.
$\text{cov(meager)}$ talks about subsets of $\mathbb{R}$, As can we code an open set of reals?
I want to prove this fact. A suggestion please.