Product topology

The least topology on a product set that makes every coordinate projection continuous. Its basic open sets restrict only finitely many coordinates.

Coordinate projection: A map from a Cartesian product to one factor that returns the corresponding coordinate. The product topology is defined as the coarsest topology making every such map continuous.
Cartesian product: The set of tuples formed by choosing one element from each of a family of sets. The product topology places a topology on this underlying set of tuples.
Box topology: A topology on a product whose basic open sets may restrict every coordinate, including infinitely many coordinates. Unlike the product topology, it permits unrestrictedly many coordinate conditions at once.
Tychonoff's theorem: The theorem that an arbitrary product of compact topological spaces is compact in the product topology. Its compactness conclusion depends on using the product rather than the box topology.
Product topology basis: A basis for the product topology consisting of products of open sets that restrict only finitely many coordinates. These basic sets generate exactly the topology required by the projection definition.
Topology: A collection of subsets of a set, containing the empty set and whole set and closed under arbitrary unions and finite intersections. Product topology is a particular topology chosen to satisfy a condition on projections.
Subspace topology: The topology on a subset inherited by intersecting that subset with open sets of an ambient space. It restricts an existing topology to a subset, rather than assembling one from projections.
Pointwise convergence: Convergence of functions at each point of their domain, considered separately. On a function space, the product topology over domain points induces pointwise convergence.
Initial topology: The coarsest topology on a set making a specified family of maps from it continuous. Product topology is the initial topology induced by its coordinate projections.
Continuous function: A function between topological spaces whose inverse image of every open set is open. Continuity of all coordinate projections determines the product topology.

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