# Is the covering property $\Omega \choose \text{T}$ closed under products?

Suppose that $(X_i)_{i\in I}$ is a family satisfying the covering property $\Omega \choose \text{T}$ (for the definition of this covering property, see this post).

Does $\prod_{i\in I} X_i$ necessarily satisfy $\Omega \choose \text{T}$?

First, if you allow $I$ to be infinite, then notice that $\mathbb{N}$ (with the discrete topology) satisfies $\binom{\Omega}{\Gamma}$ (and thus your property), but the Baire space $\mathbb{N}^\mathbb{N}$ does not satisfy Menger's property $\mathsf{S}_\mathrm{fin}(\mathrm{O},\mathrm{O})$, which is implied by your property.
For a product of two spaces, the answer remains "No," by the same method: Todorcevic proved there are (in ZFC) two $\binom{\Omega}{\Gamma}$ spaces whose product is not Lindel\"of, a property weaker than yours. If you wish to restrict to very nice spaces, like real sets, then assuming the Continuum Hypothesis there are two $\binom{\Omega}{\Gamma}$ real sets whose product is not $\mathsf{S}_\mathrm{fin}(\mathrm{O},\mathrm{O})$, see Arnold W. Miller, Boaz Tsaban, Lyubomyr Zdomskyy, Selective covering properties of product spaces, II: $\gamma$~spaces (to appear in TAMS).