Question #243960
Which of the following sets Sj , (j = 1, . . . , 4) are NOT a neighbor-
hood of the given point a.
(1) S1 = (−2, 3), a = 1; (2) S2 = [−3, ∞), a = 2.
(3) S3 = {x: |x − 1| ≤ 2} , a = 3; (4) S4 = {x: |x − 3| ≥ 2} , a = 1.
1
Expert's answer
2021-09-30T02:16:34-0400

ANSWER: S3,S4S_3,S_4 are NOT a neighborhood of the given point a.

EXPLANATION.

Let aSa\in S and there exist ε>0\varepsilon >0 such that (aε,a+ ε)S\left( a-\varepsilon ,a+\ \varepsilon \right) \subseteq S . By the definition, SS is a neighborhood of the point a.

Hence:

(1) {1}(2,3)=S1\left\{ 1 \right\} \in (-2,3)={ S }_{ 1 } . Let ε=1,ε=1, (a1,a+1)=(11,1+1)=(0,2)(2,3)=S1\left( a-1,a+1 \right) =(1-1,1+1)=(0,2)\subset (-2,3)={ S }_{ 1 }\quad .S1S_1 is a neighborhood of the point a=1.

(2) {2}[3,)=S2\left\{ 2 \right\} \in [-3,\infty)={ S }_{ 2 } . Let ε=1,ε=1, (a1,a+1)=(21,2+1)=(1,3)[3,)=S2\left( a-1,a+1 \right) =(2-1,2+1)=(1,3)\subset [-3,\infty )={ S }_{ 2 }\quad .S2S_2 is a neighborhood of the point a=2.

(3) {3}S3={x:x12}={x:2x1 2}={x:1x 3}{ \left\{ 3 \right\} \in S }_{ 3 }=\left\{ x:|x-1|\le 2 \right\} =\left\{ x:-2\le x-1\ \le 2 \right\} =\left\{ x:-1\le x \ \le 3 \right\} .

For all ε>0(aε,a+ε) =(3ε,3+ε)(3 ,3+ε)  ,(3 ,3+ε) S3=\varepsilon >0\quad \left( a-\varepsilon ,a+\varepsilon \right) \ =(3-\varepsilon ,3+\varepsilon )\supset (3\ ,3+\varepsilon )\ \ , (3\ ,3+\varepsilon )\cap \ { S }_{ 3 }=\emptyset . So, S3S_3 is not a neighborhood of the point a=3.

(4) {1}S4={x:x3 2}={x:  x32  }{x:2x3  }={x:  x1  }{x:x5   }.{ \left\{ 1 \right\} \in S }_{ 4 }=\left\{ x:|x-3\ |\ge 2 \right\} =\left\{ x:\ \ x-3\le -2\ \ \right\} \cup \left\{ x:2\le x-3\ \ \right\} =\left\{ x:\ \ x\quad \le 1\ \ \right\} \cup \left\{ x:x\ge 5\ \ \ \right\} . For all ε>0 (aε,a+ε) =(1ε,1+ε)(1 ,1+ε)\varepsilon >0\ \left( a-\varepsilon ,a+\varepsilon \right) \ =(1-\varepsilon ,1+\varepsilon )\supset (1\ ,1+\varepsilon )\quad and (1 ,1+ε)(1,5)(1\ ,1+\varepsilon )\cap \left( 1,5 \right) is not an empty set. Therefore, (1 ,1+ε)(1\ ,1+\varepsilon ) is not completely contained in S4S_4 . So, S4S_4 is not a neighborhood of the point a=1.


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