Show that \(\mathbb{R}^J\) is a Baire space in the box, product, and uniform topologies.
Solution
For the product topology, let \(G_1, G_2, \ldots\) be dense open sets and \(U_0\) a nonempty basic open set. Recursively choose nonempty basic opens \(U_n\) with \[ \overline{U_n}\subset U_{n-1}\cap G_n, \] changing only finitely many coordinates at stage \(n\) and choosing closed bounded coordinate intervals there. In each coordinate the resulting nested closed intervals have the finite intersection property; choosing one point in every coordinate gives a point of \(U_0\cap\bigcap G_n\).
For the box topology, perform the same construction coordinatewise, now shrinking every coordinate interval at stage \(n\) and arranging diameter \(<1/n\). Completeness of \(\mathbb R\) gives a point in every coordinate intersection, hence a point of \(U_0\cap\bigcap G_n\).
For the uniform topology, the uniform metric \[ \bar\rho(x,y)=\sup_{j\in J}\min\{|x_j-y_j|,1\} \] is complete: a Cauchy sequence is uniformly Cauchy in every coordinate, has coordinatewise limits, and the same uniform estimate gives convergence in \(\bar\rho\). Hence the uniform topology is Baire by the complete-metric Baire theorem.
Thus \(\mathbb R^J\) is Baire in all three topologies.