Not meager in $V^{\mathbb{C}_I}$

Question

Assume $A\subseteq 2^\omega $ is not meager in any non-empty open set, in The ground model $V $. Then is not meager in any non-empty open set, in $ V^{\mathbb{C}_I}$ where $\mathbb{C}_I$ is Cohen forcing.

Any ideas, thanks.

Answer 1

This is a standard Fubini type argument for category. Letting B to be a Cohen name for an F-sigma meager set, consider the set W of all pairs (x, y) of reals such that some initial segment of x forces y to be in B. Check that W is a meager borel set (use Kuratowski-Ulam theorem which is Fubini theorem for category). Hence if A is any non meager set in V, then for some y in A, $W^y = \{ x: (x,y) \in W \}$ is meager. It follows that A cannot be covered by B in any Cohen extension of V.