Definition 1

Given a family $(X_i)_{i \in I}$ of topological spaces, the product space of this family is the product set $X = \prod_{i \in I} X_i$ with the topology which is the product of the topologies of the $X_i$ (§ 2, no. 3, Example 3). The spaces $X_i (i \in I)$ are called the factors of X.

By virtue of § 2, no. 3, Proposition 4, the product topology on $X$ has as a base the set $\mathfrak{B}$ of finite intersections of sets of the form $\overline{\mathrm{pr}}_i^{-1}(U_i)$, where $U_i$ is open in $X_i$; these sets are products $\prod_{i \in I} A_i$, where $A_i$ is open in $X_i$ for each $i \in I$ and $A_i = X_i$ for all but a finite number of indices. These sets will be called elementary sets.

If $\mathfrak{B}_i$ is a base of the topology of $X_i$ (for each $i \in I$), it is clear that the elementary sets $\prod_{i \in I} A_i$ such that $A_i \in \mathfrak{B}_i$ for each index $i$ such that $A_i \neq X_i$ form another base of the product topology. The elementary sets of this type which contain a given point $x \in X$ thus form a fundamental system of neighbourhoods of $x$ ($§ 1$, no. 3, Proposition 3).

If $I$ is a finite set, the construction of the product topology from the topologies of the factors $X_i$ is simpler: the elementary sets are just products $\prod_{i \in I} A_i$, where $A_i$ is any open subset of $X_i$, for each $i \in I$ (cf. Exercise 9).