Let and be metric spaces and let be a continuous surjective map such that for evey .
a)If is complete, is complete?
b)If is complete, is complete?
Will you please tell us about the source of the problems that you have posted?
So far as I can see, you have made no effort at any solutions whatsoever.
Are these from a list of prelims for your degree program?
These questions are so wide ranging, I must ask you to explain your question
yes i am studying for the prelim and these problems are from old prelims like 5 to 10 years ago. i am trying to get help from a lot of people to understand these problems. and also i am trying to solve them myself. like this one i posted , i think i got the second part.
Assume that is complete.
Let be Caushy in .And using the fact that is const, for there exists such that implies such that . so is also Caushy in .
Since is complete, . Since f is surjective, there exists such that .
Since converges to , for , there exists such that implies . Since , . So converges to . Thus is complete.
I am stuck in part 1 so if you can help me with this, i am really greatful. Thank you.