The uniform metric on product spaces

If J is a set and Xj are topological spaces for each j ∈ J , let X = ∏ j∈J Xj and let πj : X → Xj be the projection maps. A basis for the product topology on X are those sets of the form ⋂ j∈J0 π −1 j (Uj), where J0 is a finite subset of J and Uj is an open subset of Xj , j ∈ J0. Equivalently, the product topology is the initial topology for the projection maps πj : X → Xj , j ∈ J , i.e. the coarsest topology on X such that each projection map is continuous. Each of the projection maps is open. The following theorem characterizes convergent 1James Munkres, Topology, second ed., p. 121, Theorem 20.1. 2John L. Kelley, General Topology, p. 90, Theorem 2.