Continuity in Infinite Product Spaces

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

  1. kehagiat

    kehagiat Guest

    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.

    kehagiat, Sep 16, 2011
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  2. kehagiat

    Gavin Wraith Guest

    In message <>
    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
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