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Math Help - two questions about a proof of a theorem on Borel spaces

  1. #1
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    two questions about a proof of a theorem on Borel spaces

    Hello everyone,
    I would really appreciate some help, as I'm stuck.

    The following is a theorem from R. M. Dudley's Real Analysis and Probability.



    First, the yellow. Theorem 13.1.5 that is mentioned states the following:

    It says that there is a continuous function from onto B, not from any complete separable metric space. Does this follow somehow?

    Then, the red. I don't see why would such neighborhoods exist.
    Why would there even exist two closed disjoint neighborhoods of and ?

    Could someone help?
    Thank you.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by harriette View Post
    Hello everyone,
    I would really appreciate some help, as I'm stuck.

    The following is a theorem from R. M. Dudley's Real Analysis and Probability.



    First, the yellow. Theorem 13.1.5 that is mentioned states the following:

    It says that there is a continuous function from onto B, not from any complete separable metric space. Does this follow somehow?

    Then, the red. I don't see why would such neighborhoods exist.
    Why would there even exist two closed disjoint neighborhoods of and ?

    Could someone help?
    Thank you.
    I am not sure I can help, take everything I can say with a grain of salt. But, isn't the red true because this A is normal?
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