Assume that for any non-empty subset U CR, if U is ajar, then f¹(U) is ajar. Prove that f is continuous everywhere. Hint: Let a € R and > 0 and consider the ajar set (f(a)- e, f(a) + c) (you may assume this set is ajar without proof). You may also prove and use these facts 1) for all non-empty BCR, f(f¹(B)) CB; 2) f(A₁) ≤ f(A₂) if non-empty sets A₁, A2 satisfies A₁ CA₂ CR. Both proofs should be very short; and 3) the results of part c).

Assume that for any non-empty subset U CR, if U is ajar, then f¹(U) is ajar. Prove that f is continuous everywhere. Hint: Let a € R and > 0 and consider the ajar set (f(a)- e, f(a) + c) (you may assume this set is ajar without proof). You may also prove and use these facts 1) for all non-empty BCR, f(f¹(B)) CB; 2) f(A₁) ≤ f(A₂) if non-empty sets A₁, A2 satisfies A₁ CA₂ CR. Both proofs should be very short; and 3) the results of part c).

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 13E: 13. Consider the set of all nonempty subsets of . Determine whether the given relation on is...

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