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Math Help - Problem on Residual Sets

  1. #1
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    Problem on Residual Sets

    Cheers everyone. I'm going through a paper by Paul Erdos and am slightly flummoxed by one of the steps in his proof. Here goes:

    Let A and B be residual sets in \mathbb{R} (in the sense of Baire category). For a fixed real number x, the sets A_x = \{x - a : a \in A\} and B_x = \{x / b : b \in B\} are also residual.

    I can see why this is true intuitively, particularly for A_x, since a simple linear transformation of the elements of A isn't going to effect the category of the complementary set, but I can't seem to drum up a rigorous proof. No doubt it is going to involve the nowhere density of related sets, but I can't seem to connect the dots. Any help would be greatly appreciated. Many thanks.
    Last edited by AlephZero; July 9th 2009 at 07:35 PM.
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  2. #2
    Super Member Rebesques's Avatar
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    Sure you mean A_x is necessarily residual?
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  3. #3
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    Quote Originally Posted by Rebesques View Post
    Sure you mean A_x is necessarily residual?
    That is what I'm trying to prove; A residual in \mathbb{R} \implies A_x residual in \mathbb{R}, and B residual in \mathbb{R} \implies B_x residual in \mathbb{R}.

    Erdos states this without proof in a paper of his, and I am trying to understand why it is true.
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