Product Topology - Definition

Definition

Given X such that

or the (possibly infinite) Cartesian product of the topological spaces X_i, indexed by, and the canonical projections p_i : X → X_i, 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 p_i 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 U_i is open in X_i and U_i ≠ X_i only finitely many times.

The product topology on X is the topology generated by sets of the form p_i−1(U), where i is in I and U is an open subset of X_i. In other words, the sets {p_i−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 p_i−1(U). The p_i−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 X_i. 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 X_i gives a basis for the product .

In general, the product of the topologies of each X_i 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.