= Convergence in a product space
{title2=$x^{(n)}\to x\iff\forall i\in I,\ \pi_i(x^{(n)})\to\pi_i(x)$}
= Coordinatewise convergence
{synonym}
For arbitrary <topological spaces> $X_i$, an <index set> $I$, a <sequence> $x^{(n)}\in\prod_iX_i$, and a point $x$ in that <Cartesian product>, <convergence in a product space> to $x$ in the <product topology> is equivalent to <coordinatewise convergence> in every factor. No countability or <Hausdorff> assumption on the factors is needed. For a <product space> of copies of the <real line>, regarding each point as a <function> $I\to\mathbb R$ identifies this with <pointwise convergence>.
Proof outline: the forward implication follows because each <projection map> is <continuous>. Conversely, given a basic <neighborhood> $B(F,U)$ of $x$, each of its finitely many coordinate tests holds for all $n\geq N_i$. Taking $N=\max_{i\in F}N_i$ makes all tests hold simultaneously; if $F$ is empty, any threshold works. This proves eventual membership in every <neighborhood>, rather than only in a sampled collection of cylinders. The same equivalence holds for <nets>: a <directed set> supplies a common upper bound for finitely many thresholds. <Sequences> alone need not characterize the whole <topology> of an arbitrary <product space>.
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