It is the usual notion of ‘product’ of topological spaces, but should be distinguished from the box product?, which is sometimes useful. (They are the same for finite products.)

The Tychonoff topology or product topology on $\prod_i {|X_i|}$ is the initial topology generated by the $\pi_i$. The Tychonoff product or topological product or simply product of the spaces $X_i$ is the set $\prod_i {|X_i|}$ equipped with the Tychonoff product topology.

$\{ \cap_{j \in J} \p_j^{-1}(U_j) | J \subset I \; finite \; subset \; and U_j \in \tau_{X_j}\}
\,,$

where $p_i \colon \prod_{j \in I} X_j \to X_i$ are the defining projection maps.

In particular if $I$ itself is already finite, then a basis of the product topology is given by the Cartesian products of elements of a basis in the factors.