= Box topology
{wiki}
The box topology on $\prod_{i\in I}X_i$ has a <basis of a topology> consisting of all <Cartesian products> $\prod_iU_i$ with every $U_i$ an <open set> in $X_i$; infinitely many coordinates may be restricted. It contains the <product topology> and agrees with it for finite $I$.<18.901, Fall 2024 lecture notes>[Lecture 12, Exercise 12.1.2(1)-(2), printed page 43: box basis, comparison with the product topology, and finite-factor equality]{ref}
For copies of the usual <real line>, the <box topology> is strictly finer when $I=\mathbb N$. The box $W=\prod_{j\geq1}(-1/2,1/2)$ is not an <open set> in the <product topology>: a basic <cylinder set> containing zero leaves some coordinate $j$ unrestricted, so it contains $e_j\notin W$. This is a construction for every finite coordinate set, not evidence from testing a finite prefix.
Back to article page
