I need help with and is due monday 3) Show the inverse image of closed sets are
ID: 1942451 • Letter: I
Question
I need help with and is due monday3) Show the inverse image of closed sets are closed
Please Help me thank you very much
Explanation / Answer
>Let f:R-->R be a continuous map, and let U be a closed subset of R. >How do I show that that pre-image of U is also closed in R? If you already know that f^{-1}[U] is open in R for all open U: C is closed --> RC is open and f^{-1}[RC] = R ^{-1}[C] is open in R, and then f^{-1}[C] is closed. Or else use sequences eg.: Let x be in cl (f^{-1}[C] ), where C is closed. Then there is a sequence x_n from f^{-1}[C] such that x_n --> x. But then f(x_n) is in C (for all n), f(x_n) --> f(x) by continuity, and thus, as C is closed we have that f(x) is in C. So x is in f^{-1}[C], and thus this set is closed too.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.