# Continuity in Infinite Product Spaces

Discussion in 'Math Research' started by kehagiat, Sep 16, 2011.

1. ### kehagiatGuest

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.

Thn

kehagiat, Sep 16, 2011

2. ### Gavin WraithGuest

In message <160920111122285423%-state.edu.invalid>
The projection map U -> U1xU2xU3x....xUn is continuous, by definition
of the product topology. Hence so is its composite with a continuous
map U -> R.

Gavin Wraith, Sep 17, 2011