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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 $\Omega \choose \text{T}$?

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The answer is "No." This is yet another instance where selection principles are useful. One uses knowledge on more understood, related properties, to answer questions on less understood ones.

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).

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