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Math Help - Various conceptual questions

  1. #1
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    Various conceptual questions

    Hi all,

    I'm working my way through a proof of the Baire Category Theorem & need help with some of the concepts involved...

    First off, can someone help me with understanding the definition of "nowhere dense" & "somewhere dense"? I've got the definition of somewhere dense as being "a set A is somewhere dense in a complete metric space B if the closure of A contains an open subset of B" ... For nowhere dense I have A is nowhere dense in B if the interior of its closure is dense in B.

    I'm comfortable with all the other terms/concepts in the definition, even "dense", but these two definitions just haven't really 'clicked' I guess. I think what I need is a few examples. The books I'm working from have none :-/ ... So yeah if someone could really spell it out for me and throw a few examples in I'd appreciate that very much.

    Also, with regards to an interior of a closure, is this interpretation:

    If A is open then the interior of it's closure is A itself ... if A is closed then the interior of it's closure is a subset of A.

    Correct?

    Next,

    Part of the BCT as it's stated in the book I'm working from is: Let (X,d) be a complete metric space. Then if U1, U2, U3... are countably many dense open subsets of X then the intersection of N Un's is dense in X.

    I only have a small photocopied part of this book, and I have been introduced to "countable" from another book as meaning either finite or countably infinite. Is it true that some authors (including this one) use it only to mean countably infinite, and NOT finite? Otherwise, where does this break down - the rationals and irrationals are both dense in the reals, but their intersection is null, and the closure of the null set is null, which is not the reals... ?

    Lastly, could someone suggest a few simple examples of the BCT?



    Thank you very much to anyone who takes the time to answer any of this!
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  2. #2
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    Quote Originally Posted by RanDom View Post
    I'm working my way through a proof of the Baire Category Theorem & need help with some of the concepts involved...

    First off, can someone help me with understanding the definition of "nowhere dense" & "somewhere dense"? I've got the definition of somewhere dense as being "a set A is somewhere dense in a complete metric space B if the closure of A contains an open subset of B" ... For nowhere dense I have A is nowhere dense in B if the interior of its closure is dense in B. No! A is nowhere dense in B if the interior of its closure is empty.

    I'm comfortable with all the other terms/concepts in the definition, even "dense", but these two definitions just haven't really 'clicked' I guess. I think what I need is a few examples. The books I'm working from have none :-/ ... So yeah if someone could really spell it out for me and throw a few examples in I'd appreciate that very much.
    A simple example of a nowhere dense subset of the real line is the set of integers. They form a closed subset with empty interior. But the rationals are everywhere dense in the reals, because the closure of the rationals is the whole line.

    A less trivial example of a nowhere dense subset of the real is the Cantor set. Again, it is closed, but its interior is empty.

    Quote Originally Posted by RanDom View Post
    Also, with regards to an interior of a closure, is this interpretation:

    If A is open then the interior of it's closure is A itself Not necessarily. For example, the set \color{red}(0,1)\cup(1,2) has closure [0,2]. The point 1 is in the interior of the closure, but is not in the original set.
    ... if A is closed then the interior of it's closure is a subset of A.

    Correct?

    Next,

    Part of the BCT as it's stated in the book I'm working from is: Let (X,d) be a complete metric space. Then if U1, U2, U3... are countably many dense open subsets of X then the intersection of N Un's is dense in X.

    I only have a small photocopied part of this book, and I have been introduced to "countable" from another book as meaning either finite or countably infinite. Is it true that some authors (including this one) use it only to mean countably infinite, and NOT finite? Otherwise, where does this break down - the rationals and irrationals are both dense in the reals, but their intersection is null, and the closure of the null set is null, which is not the reals... ?
    But neither the rationals nor the irrationals form open sets, so the theorem does not apply. In fact, the BCT applies just as well to finite collections of subsets as it does to countably infinite collections.

    Quote Originally Posted by RanDom View Post
    Lastly, could someone suggest a few simple examples of the BCT?
    It is an ingredient of the standard proofs of the three big basic theorems of functional analysis, the Uniform Boundedness, Open Mapping and Closed Graph theorems.
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    Great, thanks!
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