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6. Below are several statements about compact sets; some are true and some are not. Prove the true ones, and give a counterexample for the false ones. (a) Let Kn be a compact set for each n E N. Then K = U Kn must be compact. n=0 (b) Let F be a compact set for each 2 E A. Then F = O Fa must be compact. λΕΛ (c) Let A be an arbitrary subset of R, and let K be compact. Then ANK is compact. (d) Let K be compact and F be closed. Then K F = {x€K|x ¢ F } is open.

6. Below are several statements about compact sets; some are true and some are not. Prove the true ones, and give a counterexample for the false ones. (a) Let Kn be a compact set for each n E N. Then K = U Kn must be compact. n=0 (b) Let F be a compact set for each 2 E A. Then F = O Fa must be compact. λΕΛ (c) Let A be an arbitrary subset of R, and let K be compact. Then ANK is compact. (d) Let K be compact and F be closed. Then K F = {x€K|x ¢ F } is open.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 17E: In each of the following parts, a relation R is defined on the power set (A) of the nonempty set A....
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pls solve all parts (i)-(iv) with the proper explanation I'll give you multiple likes
6. Below are several statements about compact sets; some are true and some are not. Prove the true ones, and give a counterexample for the false ones. (a) Let Kn be a compact set for each n E N. Then K = |U Kn must be compact. n=0 (b) Let F, be a compact set for each 2 E A. Then F = O F, must be compact. (c) Let A be an arbitrary subset of R, and let K be compact. Then ANK is compact. (d) Let K be compact and F be closed. Then K F = {x€K|x ¢F} is open.
Transcribed Image Text:6. Below are several statements about compact sets; some are true and some are not. Prove the true ones, and give a counterexample for the false ones. (a) Let Kn be a compact set for each n E N. Then K = |U Kn must be compact. n=0 (b) Let F, be a compact set for each 2 E A. Then F = O F, must be compact. (c) Let A be an arbitrary subset of R, and let K be compact. Then ANK is compact. (d) Let K be compact and F be closed. Then K F = {x€K|x ¢F} is open.
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