neighborhood of a set A

• Oct 1st 2009, 11:12 AM
ElieWiesel
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 $\displaystyle G_{\delta}$
• Oct 1st 2009, 12:10 PM
Plato
Quote:

Originally Posted by ElieWiesel
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) $\displaystyle \bigcap_{n\in \mathbb{Z}^+} N_{\frac{1}{n}} (A)$?