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Math Help - neighborhood of a set A

  1. #1
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    neighborhood of a set A

    Define the  \epsilon neighborhood of a set A by:


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

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

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

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

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

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

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

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

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