The product topology is the topology of coordinatewise observation. It is deliberately weaker than the box topology in infinite products, and that weakness is exactly what makes compactness and universal properties work.
Finite and infinite products
Basic product-open sets restrict only finitely many coordinates.
Definition. For spaces \(\{X_i:i\in I\}\), the product topology on \(\prod_{i\in I}X_i\) has subbasis \(\pi_i^{-1}(U)\), where \(U\subseteq X_i\) is open and \(\pi_i\) is the coordinate projection. A basis consists of sets \(\prod_i U_i\) where each \(U_i\) is open and \(U_i=X_i\) for all but finitely many \(i\).
Proposition. Each projection \(\pi_i:\prod_jX_j\to X_i\) is continuous, and the product topology is the coarsest topology with this property.
Proof. The subbasic sets are exactly inverse images of open sets under projections, so projections are continuous. Any topology making all projections continuous must contain every \(\pi_i^{-1}(U)\), hence must contain the topology generated by these subbasic sets.
Universal property
Maps into products are continuous precisely when their coordinate maps are continuous.
Theorem. For a map \(f:Y\to\prod_{i\in I}X_i\), \(f\) is continuous if and only if every coordinate map \(\pi_i\circ f:Y\to X_i\) is continuous.
Proof. If \(f\) is continuous, then each coordinate map is a composite of continuous maps. Conversely, it is enough to check inverse images of subbasic sets: \(f^{-1}(\pi_i^{-1}(U))=(\pi_i\circ f)^{-1}(U)\), which is open by continuity of the coordinate map.
Remark. This theorem is often the cleanest way to prove continuity of curves, functions, or random elements with values in a product space.
Box topology comparison
The box topology allows infinitely many coordinate restrictions and is usually too fine for analysis.
Definition. The box topology on \(\prod_iX_i\) has basis all boxes \(\prod_iU_i\) with each \(U_i\subseteq X_i\) open, with no finite-support restriction.
Proposition. For infinite products, the box topology is finer than the product topology, and the inclusion can be strict.
Proof. Every product-basic set is a box-basic set, so the box topology is finer. In \(\mathbb R^{\mathbb N}\), the box \(\prod_n(-1/n,1/n)\) is box-open but not product-open, because every product-basic neighborhood of \(0\) leaves all but finitely many coordinates unrestricted.
Products and separation
Many separation axioms are stable under products because they can be tested coordinatewise.
Proposition. A product \(\prod_iX_i\) is Hausdorff if and only if every factor \(X_i\) is Hausdorff.
Proof. If the product is Hausdorff, each factor embeds into the product after choosing basepoints in the other coordinates, so each factor is Hausdorff. Conversely, if two product points differ, they differ in some coordinate \(i\). Separate those two coordinates by disjoint open sets in \(X_i\), and pull them back by \(\pi_i\) to disjoint open neighborhoods in the product.
Corollary. Products of \(T_1\) spaces are \(T_1\), and products of regular spaces are regular.
Proof. The \(T_1\) case follows by the same coordinate argument. Regularity follows by separating a point from a basic closed neighborhood using finitely many regular separations in the relevant coordinates, then intersecting the corresponding pulled-back neighborhoods.
Product versus box topology
Counterexample. On \(\mathbb R^{\mathbb N}\), the product topology has basic neighborhoods \(\prod_nU_n\) with \(U_n=\mathbb R\) for all but finitely many \(n\). The box topology allows every \(U_n\) to be restricted. Hence \[ (-1,1)^{\mathbb N} \] is box-open but not product-open. The sequence \[ x^{(k)}=(0,\ldots,0,1,1,\ldots) \] with the first \(k\) coordinates zero converges coordinatewise, hence in the product topology, but never enters this box neighborhood of zero.
The product topology is the coarsest topology making all projections \(\pi_i\) continuous. Consequently a map \[ f:X\to\prod_iY_i \] is continuous exactly when every coordinate \(\pi_i\circ f\) is continuous. This universal property, rather than visual resemblance to finite Cartesian products, determines the correct infinite construction.