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Products of spaces

Given X such that

is the Cartesian product of the topological spaces X, indexed by i ∈ I {\displaystyle i\in I}, and the canonical projections p : X → X, the product topology on X is defined as the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections p are continuous. The product topology is sometimes called the Tychonoff topology.

The open sets in the product topology are unions (finite or infinite) of sets of the form ∏ i ∈ I U i {\displaystyle \prod _{i\in I}U_{i}}, where each U is open in X and U ≠ X only finitely many times. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the X gives a basis for the product ∏ i ∈ I X i {\displaystyle \prod _{i\in I}X_{i}}.

The product topology on X is the topology generated by sets of the form p (U), where i is in I and U is an open subset of X. In other words, the sets { p (U)} form a subbase for the topology on X. A subset of X is open if and only if it is a (possibly infinite) union of intersections of finitely many sets of the form p (U). The p (U) are sometimes called open cylinders, and their intersections are cylinder sets.

In general, the product of the topologies of each X forms a basis for what is called the box topology on X. In general, the box topology is finer than the product topology, but for finite products they coincide.

Related to compactness is Tychonoff's theorem: the (arbitrary) product of compact spaces is compact.