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Topic: Continuity in Infinite Product Spaces
Replies: 1   Last Post: Sep 17, 2011 10:30 AM

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Posts: 3
Registered: 9/16/11
Continuity in Infinite Product Spaces
Posted: Sep 16, 2011 1:22 PM
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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.


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