= Finite-coordinate basis for the product topology
{title2=$B(F,U)=\bigcap_{i\in F}\pi_i^{-1}(U_i)$}
For a <finite set> $F\subseteq I$ and <open sets> $U_i\subseteq X_i$ for $i\in F$, the sets
$$
B(F,U)=\bigcap_{i\in F}\pi_i^{-1}(U_i)
=\prod_{i\in I}V_i,\qquad
V_i=\begin{cases}U_i&i\in F,\\X_i&i\notin F\end{cases}
$$
form a <basis of a topology> for the <product topology>. The empty <set intersection> is $X$. Each basic <open set> is a <cylinder set>; all coordinates outside $F$ are free. Arbitrary <open sets> are unions of these basic tests and need not themselves depend on one fixed finite collection of coordinates.<Point-Set Topology by Romyar Sharifi/Point-Set Topology, Chapter 2: Continuous functions>[Proposition 2.2.1, Definition 2.2.2 and Remark 2.2.8: finite-coordinate basis and general open subsets]{ref}
Why finitely many? Any <topology> making the <projection maps> <continuous> must contain their <preimages> of <open sets>, and hence finite <set intersections> of those <preimages>. Taking arbitrary unions supplies exactly the <product topology>. Intersecting two basic <cylinder sets> still restricts only the finite union of their coordinate sets.
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