I'm trying to prove that a countably infinite cartesian product of connected spaces is connected. Let X be a connected space and let Y be the countably infinite cartesian product of copies of X, and suppose Y is equipped with the product topology. So suppose Y is not connected. Let A and B be two disjoint, nonempty, open subsets of Y s.t. A U B = Y. Since A and B are open sets in the product topology, they are both of the form U1 X U2 X .... where each Ui's are not X for only finitely many i and each Ui is open in X. So let k be the sup of the index's for A and B which Ui's is not a copy of X. So A intersect B = (U1 intersect Y1) X ... X (Uk intersect YK) x X x X... = empty set. Hence for some j less than k, Uj intersect Wj is empty. Hence Uj and Wj form a seperation of X since A U B is Y, then Ui U Wi = X, and for Uj and Wj, they are disjoint and nonempty. So where did I go wrong?