For sets A and B, define A+B = {a + b: a € A, b E B}. (a) Prove that if A and B are open, then A+ B is open. (b) Prove that if A and B are compact, then A+ B is compact. (c) It is not true that the sum of closed sets must be closed. Provide an example to demonstrate this.

For sets A and B, define A+B = {a + b: a € A, b E B}. (a) Prove that if A and B are open, then A+ B is open. (b) Prove that if A and B are compact, then A+ B is compact. (c) It is not true that the sum of closed sets must be closed. Provide an example to demonstrate this.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.4: Prime Factors And Greatest Common Divisor

Problem 17E

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Can you please answer all three parts simply, or just the last two (b,c). Either way, Im just looking for a simple soultion for a better understanding. Thank you.

For sets A and B, define A+ B = {a +b: a € A, b e B}.
(a) Prove that if A and B are open, then A+ B is open.
(b) Prove that if A and B are compact, then A+ B is compact.
(c) It is not true that the sum of closed sets must be closed. Provide an example to demonstrate
this.

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