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Math Help - Nowhere dense in a metric space

  1. #1
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    Nowhere dense in a metric space

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    A subset A of a metric space X is nowhere dense in X if and only if each non-empty open set in X contains an open ball whose closure is disjoint from A.


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    Some definitions
    1. A subset A of a metric space X is nowhere dense in X if \bar{A} has empty interior.
    2. Let A be a subset of a metric space X. A point x in A is an interior point of A provided that there is an open set O which contains x and is contained in A.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by aliceinwonderland View Post
    Show that

    A subset A of a metric space X is nowhere dense in X if and only if each non-empty open set in X contains an open ball whose closure is disjoint from A.


    -------------------------------------------------------------
    Some definitions
    1. A subset A of a metric space X is nowhere dense in X if \bar{A} has empty interior.
    2. Let A be a subset of a metric space X. A point x in A is an interior point of A provided that there is an open set O which contains x and is contained in A.
    NOTE: You may want to wait for a more senior knowledgable member to either validate or correct my suggestion.

    This is almost by defintion. Supose the opposite that A is dense in X and that the hypothesis holds...use this to show that not every point of X is a limit point of A
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  3. #3
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    Quote Originally Posted by aliceinwonderland View Post
    A subset A of a metric space X is nowhere dense in X if and only if each non-empty open set in X contains an open ball whose closure is disjoint from A.
    Is it still true if the word "closure" is taken out such that
    "A subset A of a metric space X is nowhere dense in X if and only if each non-empty open set in X contains an open ball which is disjoint from A"
    If it is wrong, any counterexample?
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