# Thread: neighborhood of a set A

1. ## 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 $G_{\delta}$

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