For any set S c R, let 5 denote the intersection of all the closed sets containing S. (a) Prove that S is a closed set.
For any set S c R, let 5 denote the intersection of all the closed sets containing S. (a) Prove that S is a closed set.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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![Chapter 3 • The Real Numbers
9. Prove the following.
(a) An accumulation point of a set S is either an interior point of S or a
boundary point of S.
(b) A boundary point of a set S is either an accumulation point of S or an
isolated point of S.
10. Prove: If x is an isolated point of a set S, then x e bd S.
11. If A is open and B is closed, prove that A\B is open and B\A is closed. *
12. Prove: For each x e R and & > 0, N*(x; E) is an open set.
13. Prove: (cl S) (int S) = bd S. *
14. Let S be a bounded infinite set and let x = sup S. Prove: Ifx e S, then xe S'.
*15. Prove: If x is an accumulation point of the set S, then every neighborhood of
x contains infinitely many points of S. ☆
16. (a) Prove: bd S = (cl S)n [cl (R\S)].
(b) Prove: bd S is a closed set.
17. Prove: S' is a closed set. *
18. Prove Theorem 3.4.17(c) and (d).
19. Suppose S is a nonempty bounded set and let m = sup S. Prove or give a
counterexample: m is a boundary point of S.
20. Prove or give a counterexample: If a set S has a maximum and a minimum,
then S is a closed set.
*21. Let A be a nonempty open subset of R and let Q be the set of rationals.
Prove that AnQ #Ø.
22. Let S and 7 be subsets of R. Prove the following.
(a) cl (cl S) = el S
(b) cl (SUT) = (cl S) U (cl T)
(c) cl (Sn T) S (cl S)n (cl T)
(d) Find an example to show that equality need not hold in part (c).
23. Let S and T be subsets of R. Prove the following.
(a) int S is an open set.
(b) int (int S) = int S
(c) int (SnT) = (int S)n (int T)
(d) (int S) U (int T) C int (SUT)
(e) Find an example to show that equality need not hold in part (d).
24. For any set SC R, let S denote the intersection of all the closed sets
containing S.
(a) Prove that § is a closed set.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2F2ec670b1-1d2f-410b-9166-e796df6416d3%2Fo94q4ua_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Chapter 3 • The Real Numbers
9. Prove the following.
(a) An accumulation point of a set S is either an interior point of S or a
boundary point of S.
(b) A boundary point of a set S is either an accumulation point of S or an
isolated point of S.
10. Prove: If x is an isolated point of a set S, then x e bd S.
11. If A is open and B is closed, prove that A\B is open and B\A is closed. *
12. Prove: For each x e R and & > 0, N*(x; E) is an open set.
13. Prove: (cl S) (int S) = bd S. *
14. Let S be a bounded infinite set and let x = sup S. Prove: Ifx e S, then xe S'.
*15. Prove: If x is an accumulation point of the set S, then every neighborhood of
x contains infinitely many points of S. ☆
16. (a) Prove: bd S = (cl S)n [cl (R\S)].
(b) Prove: bd S is a closed set.
17. Prove: S' is a closed set. *
18. Prove Theorem 3.4.17(c) and (d).
19. Suppose S is a nonempty bounded set and let m = sup S. Prove or give a
counterexample: m is a boundary point of S.
20. Prove or give a counterexample: If a set S has a maximum and a minimum,
then S is a closed set.
*21. Let A be a nonempty open subset of R and let Q be the set of rationals.
Prove that AnQ #Ø.
22. Let S and 7 be subsets of R. Prove the following.
(a) cl (cl S) = el S
(b) cl (SUT) = (cl S) U (cl T)
(c) cl (Sn T) S (cl S)n (cl T)
(d) Find an example to show that equality need not hold in part (c).
23. Let S and T be subsets of R. Prove the following.
(a) int S is an open set.
(b) int (int S) = int S
(c) int (SnT) = (int S)n (int T)
(d) (int S) U (int T) C int (SUT)
(e) Find an example to show that equality need not hold in part (d).
24. For any set SC R, let S denote the intersection of all the closed sets
containing S.
(a) Prove that § is a closed set.
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