Here is my question. Excuse its possible simple-mindedness.
Say I have topological spaces U1, U2, U3, ... and I form the countably infinite product space U1xU2xU3x.... .
I also have a function f:U ->R. So nominally it is a function of infinitely many variables f(u1,u2,...). But in fact it only depends on n variables, say f(u1,u2,...,un). Also it is a continuous function of these n variables.
QUESTION: is f also continuous over U ? (wrt product topology).
It seems obvious to me that it is but then I get doubts. I tried to prove it but no luck. So I will be very grateful if anybody can help.