= Continuity into a product space
{title2=$f:Y\to\prod_iX_i\text{ continuous}\iff\forall i,\ \pi_i\circ f\text{ continuous}$}
For a <topological space> $Y$ and <topological spaces> $X_i$, a <function> $f:Y\to\prod_iX_i$ is <continuous> into the <product topology> exactly when every coordinate <function> $f_i=\pi_i\circ f$ is <continuous>.<18.901, Fall 2024 lecture notes>[Lecture 5, Proposition 5.3.2, printed page 21: continuity into a product is equivalent to continuity of all component functions]{ref}
Proof outline: one direction uses <function composition> with the <continuous> <projection maps>. For the other, the <preimage> of a basic <cylinder set> is
$$
f^{-1}(B(F,U))=\bigcap_{i\in F}f_i^{-1}(U_i),
$$
a finite <set intersection> of <open sets> in $Y$. This gives <continuity> on every basic test. It explains, for example, why $t\mapsto(t,t^2,t^3,\ldots)$ from the <real line> to $\mathbb R^{\mathbb N}$ is <continuous> in the <product topology>: each coordinate is a <polynomial> <function>. The general coordinate criterion fails for the <box topology>, where infinitely many <preimages> can be required at once. For example, $g(t)=(t,t,\ldots)$ has <continuous> coordinate <functions>, but the <preimage> of the box $\prod_{j\geq1}(-1/j,1/j)$ is $\{0\}$, which is not an <open set> in the <real line>.
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