Definition
Given X such that
or the (possibly infinite) Cartesian product of the topological spaces Xi, indexed by, 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, where each Ui is open in Xi and Ui ≠ Xi only finitely many times.
The product topology on X is the topology generated by sets of the form pi−1(U), where i is in I and U is an open subset of Xi. In other words, the sets {pi−1(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 pi−1(U). The pi−1(U) are sometimes called open cylinders, and their intersections are cylinder sets.
We can describe a basis for the product topology using bases of the constituting spaces Xi. A basis consists of sets, where for cofinitely many (all but finitely many) i, (it's the whole space), and otherwise it's a basic open set of .
In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the Xi gives a basis for the product .
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.
Read more about this topic: Product Topology
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