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Thread: neighborhood of a set A

  1. #1
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    neighborhood of a set A

    Define the $\displaystyle \epsilon $ neighborhood of a set A by:


    $\displaystyle N_{\epsilon} (A) = \{ x \in M : \exists y \in A $ such that $\displaystyle d(x,y) < \epsilon \} $

    (that is, it is the collection of points in M wich are with $\displaystyle \epsilon $ of some point in A.

    (a) prove $\displaystyle N_{\epsilon} (A) $ is open.

    (b) $\displaystyle \bigcap_{\epsilon > 0} N_{\epsilon} (A) = \bar{A} $, the closure of A

    (c) A subset B $\displaystyle \subset $ M is said to be a $\displaystyle G_{\delta} $ if it is the countable intersection of open sets. Modify part (b) to prove: every closed set is a $\displaystyle G_{\delta} $
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  2. #2
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    Quote Originally Posted by ElieWiesel View Post
    Define the $\displaystyle \epsilon $ neighborhood of a set A by: $\displaystyle N_{\epsilon} (A) = \{ x \in M : \exists y \in A $ such that $\displaystyle d(x,y) < \epsilon \} $ (that is, it is the collection of points in M wich are with $\displaystyle \epsilon $ of some point in A.

    (a) prove $\displaystyle N_{\epsilon} (A) $ is open.

    (b) $\displaystyle \bigcap_{\epsilon > 0} N_{\epsilon} (A) = \bar{A} $, the closure of A

    (c) A subset B $\displaystyle \subset $ M is said to be a $\displaystyle G_{\delta} $ if it is the countable intersection of open sets. Modify part (b) to prove: every closed set is a $\displaystyle G_{\delta} $
    a) If $\displaystyle x\in N_{\epsilon} (A) $ then $\displaystyle \left( {\exists y \in A} \right)\left[ {x \in d(x,y) < \epsilon } \right]$.
    Now let $\displaystyle \delta = \min \left\{ {d(x,y),\epsilon - d(x,y)} \right\}$.
    Show that $\displaystyle \mathcal{B}(x;\delta ) \subset N_\epsilon (A)$

    c) What can you say about (b) $\displaystyle \bigcap_{n\in \mathbb{Z}^+} N_{\frac{1}{n}} (A)$?
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