Let \(X=Y=\mathbb Z\), each with the finite-complement topology. Let \(\mathcal T_{prod}\) be the product topology on \(X\times Y\), and let \(\mathcal T_{fc}\) be the finite-complement topology on \(X\times Y\).
Determine whether \[ \mathcal T_{prod}=\mathcal T_{fc}. \] If equality fails, determine which, if either, of the containments \[ \mathcal T_{prod}\subseteq\mathcal T_{fc}, \qquad \mathcal T_{fc}\subseteq\mathcal T_{prod} \] holds, and prove your answer.