ANSWER: S3,S4 are NOT a neighborhood of the given point a.
EXPLANATION.
Let a∈S and there exist ε>0 such that (a−ε,a+ ε)⊆S . By the definition, S is a neighborhood of the point a.
Hence:
(1) {1}∈(−2,3)=S1 . Let ε=1, (a−1,a+1)=(1−1,1+1)=(0,2)⊂(−2,3)=S1 .S1 is a neighborhood of the point a=1.
(2) {2}∈[−3,∞)=S2 . Let ε=1, (a−1,a+1)=(2−1,2+1)=(1,3)⊂[−3,∞)=S2 .S2 is a neighborhood of the point a=2.
(3) {3}∈S3={x:∣x−1∣≤2}={x:−2≤x−1 ≤2}={x:−1≤x ≤3} .
For all ε>0(a−ε,a+ε) =(3−ε,3+ε)⊃(3 ,3+ε) ,(3 ,3+ε)∩ S3=∅ . So, S3 is not a neighborhood of the point a=3.
(4) {1}∈S4={x:∣x−3 ∣≥2}={x: x−3≤−2 }∪{x:2≤x−3 }={x: x≤1 }∪{x:x≥5 }. For all ε>0 (a−ε,a+ε) =(1−ε,1+ε)⊃(1 ,1+ε) and (1 ,1+ε)∩(1,5) is not an empty set. Therefore, (1 ,1+ε) is not completely contained in S4 . So, S4 is not a neighborhood of the point a=1.
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