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Good evening. Here are two citations from a book about basic principles of analysis:
Цитата:
Exercise 1. A function is continuous if and only if the preimage of every open set is open.
Цитата:
Exercise 2. If is continuous on a compact set , then is compact.
I know that exercise 1 can help to prove exercise 2. I am starting (perhaps hastily) by defining . Then what allows to apply exercise 1 to function ( but )? I am even inclined to reprove exercise 1 for .
Oleg Zubelevich
Re: continuity and open sets
14.03.2012, 07:59
Последний раз редактировалось Oleg Zubelevich 14.03.2012, 08:37, всего редактировалось 1 раз.
cover with an open covering, look at the preimage of that covering
gefest_md
Re: continuity and open sets
16.03.2012, 17:18
I am going to look at some collection . But as an amateur, I can not understand the usefulness of Exercise 1, with in it, when I had to infer that each of is also open. With Exercise 1 in hand, why Exercise 2 can be addressed on restricted domain?
Oleg Zubelevich
Re: continuity and open sets
16.03.2012, 18:54
Последний раз редактировалось Oleg Zubelevich 16.03.2012, 18:55, всего редактировалось 2 раз(а).
Actually the both assertions hold true not only for the case but for general topological spaces. I believe you should use a proper course of analysis L Schwatz, for example