Answer to Question #322810 in Real Analysis for Talifhani

Question

Let P = (3, 4), ò= (0, 0), P0 = (5,0) be points in R²

equipped with the

Euclidean metric. Let R be the ring given by R ={v∈ R²: 4 < d(ò, v) ≤ 7}.

Which of the following are neighborhoods of P?

R, B1 = B(ò, 4), B2 = B(ò, 6), B3 = R \ B2, B4 = B(ò, 1) ∪ B2, B5 = B(P0,√11).

Answer 1

$\mathbb{R} ^2\,\,is\,\,a\,\,neighborhood\,\,of\,\,P, \sin ce\,\,P\in \mathbb{R} ^2,\mathbb{R} ^2\,\,is\,\,open\\B_1=B\left( O,4 \right) \,\,is\,\,not\,\,a\,\,neighborhood\,\,of\,\,P: \left\| \left( 3,4 \right) -\left( 0,0 \right) \right\| =5>4\Rightarrow \left( 3,4 \right) \notin B_1\\B_2=B\left( O,6 \right) \,\,is\,\,a\,\,neighborhood\,\,of\,\,P: \left\| \left( 3,4 \right) -\left( 0,0 \right) \right\| =5<6\Rightarrow \left( 3,4 \right) \in B_2,B_2\,\,is\,\,open\\B_3=\mathbb{R} ^2\setminus B_2\,\,is\,\,not\,\,a\,\,neighborhood\,\,of\,\,P: P\in B_2\Rightarrow P\notin B_3\\B_4=B\left( O,1 \right) \cup B_2=B_2\,\,is\,\,a\,\,neighborhood\,\,of\,\,P\,\,\left( see\,\,above \right) \\B_5=B\left( P_0,\sqrt{11} \right) \,\,is\,\,not\,\,a\,\,neighborhood\,\,of\,\,P: \left\| \left( 3,4 \right) -\left( 5,0 \right) \right\| =\sqrt{2^2+4^2}>\sqrt{11}\Rightarrow \left( 3,4 \right) \notin B_5$