4. Let f: RR. For each subset A CR, we define its preimage by f-¹(A) = { € R" | f(x) = A}. Prove that f is continuous if and only if for every open set OCR, f-¹(0) is open.
4. Let f: RR. For each subset A CR, we define its preimage by f-¹(A) = { € R" | f(x) = A}. Prove that f is continuous if and only if for every open set OCR, f-¹(0) is open.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.5: Permutations And Inverses
Problem 4E: 4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .
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Transcribed Image Text:4.
Let f: RR. For each subset ACR, we define its preimage
by f-¹(A) = { € R" | f(x) = A}.
Prove that f is continuous if and only if for every open set OCR,
f-¹(0) is open.
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