Product topology

In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is 'correct' in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; this is the sense in which the product topology is 'natural'. In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is 'correct' in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; this is the sense in which the product topology is 'natural'. Given X such that is the Cartesian product of the topological spaces Xi, indexed by i ∈ I {displaystyle iin I} , and the canonical projections pi : X → Xi, the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections pi 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 _{iin I}U_{i}} , where each Ui is open in Xi and Ui ≠ Xi for only finitely many i. In particular, for a finite product (in particular, for the product of two topological spaces), the set of all Cartesian products between one basis element from each Xi gives a basis for the product topology of ∏ i ∈ I X i {displaystyle prod _{iin I}X_{i}} . That is, for a finite product, the set of all ∏ i ∈ I U i {displaystyle prod _{iin I}U_{i}} , where U i {displaystyle U_{i}} is an element of the (chosen) basis of X i {displaystyle X_{i}} , is a basis for the product topology of ∏ i ∈ I X i {displaystyle prod _{iin I}X_{i}} . The product topology on X is the topology generated by sets of the form pi−1(Ui), where i is in I and Ui is an open subset of Xi. In other words, the sets {pi−1(Ui)} 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 pi−1(Ui). The pi−1(Ui) are sometimes called open cylinders, and their intersections are cylinder sets. In general, the product of the topologies of each Xi 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. If one starts with the standard topology on the real line R and defines a topology on the product of n copies of R in this fashion, one obtains the ordinary Euclidean topology on Rn. The Cantor set is homeomorphic to the product of countably many copies of the discrete space {0,1} and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where again each copy carries the discrete topology. Several additional examples are given in the article on the initial topology.