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Exam 2024-01-13 Question 6.ii

Exam 2024-01-13 Question 6.ii

av Sofie Angere -
Antal svar: 3

Hi, another question from this exam that I haven't managed to sort out on my own: according to the hints for the exam, 6.ii is false, which means that there is some closed set K \subseteq \mathbb{R} such that f^{-1}(K) is not a closed set. But as far as I can see, f(x) = 1/(2+x^2) is continuous, so shouldn't the inverse image of every closed set be closed because of that?


Thanks,

Sofie

Som svar till Sofie Angere

Sv: Exam 2024-01-13 Question 6.ii

av Kilian Liebe -
You're correct, the function is continuous and therefore the preimage of any closed set should be closed. I'm not sure what hint you're referring to, I don't see any hints in the exam.
Som svar till Kilian Liebe

Re: Sv: Exam 2024-01-13 Question 6.ii

av Sofie Angere -
Hi,

it's in the solutions, in the file 240113s.pdf - they're called "hints for solutions there", on the last page. That one claims that question 6.ii is false.
Som svar till Sofie Angere

Sv: Re: Sv: Exam 2024-01-13 Question 6.ii

av Kilian Liebe -
I see now, I didn't scroll down far enough. It has to be incorrect then, because the preimage of a closed set under a continuous map is always closed (this is even sometimes used as a definition for continuity).

Perhaps the person writing the answers was thinking about compact sets, because then the statement becomes false (there are compact sets K such that f^{-1}(K) is not compact for the function f in the question).