Product Topology

Let $I$ be a set and consider the collection $C=\{(X_i,\tau_i)\}_{i \in I}$ of topological spaces. Consider a topological space $\left( \displaystyle\prod_{i \in I} X_i, \tau \right)$ and an arbitrary $Z \in \tau$. By construction, there is a collection $\{Z_i\}_{i \in I}$ such that each $Z_i \in \tau_i$ and $Z = \displaystyle\prod_{i \in I} Z_i$. We say that $\tau$ is the product topology associated to $C$ provided that only finitely many of the open sets $Z_i$ are not equal to $X_i$.