= Product topology
{wiki}
The product topology turns separate observations into a notion of closeness: a basic test checks finitely many coordinates, with a separate tolerance in each. For an <index set> $I$ and <topological spaces> $X_i$, the underlying <Cartesian product> and its coordinate <projection maps> are
$$
X=\prod_{i\in I}X_i=\{(x_i)_{i\in I}:x_i\in X_i\},\qquad
\pi_i:X\to X_i,\quad\pi_i(x)=x_i.
$$
The product topology is the coarsest <topology> making every $\pi_i$ a <continuous map>.<Point-Set Topology by Romyar Sharifi/Point-Set Topology, Chapter 2: Continuous functions>[Definition 2.2.4 and Proposition 2.2.5: coordinate projections and the coarsest topology making them continuous]{ref}
The <Cartesian product> specifies which points exist; the <product topology> specifies their <neighborhoods>. Finite observations motivate its <finite-coordinate basis for the product topology>, while <convergence in a product space> and <continuity into a product space> explain why it is useful. The <compactness> result for this construction is <Tychonoff's theorem>.
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