Here are two more questions that have been bothering me. for the first, i don't think my solution is good enough, there's somethign too "obvious" about it. for the second, i haven't a clue
1) Let and be metric spaces. Assume that is complete and that there exists which is bijective and continuous. Suppose that is uniformly continuous. Prove that is complete
2) Prove that any open subset of a complete metric space is homeomorphic to a complete metric space. (Hint: If is the complement of the open set, use the metric
Here are definitions/theorems i figured would come in handy:
Defintion: A metric space is complete if every Cauchy sequence in is convergent to a point of .
Theorem: Any convergent sequence in a metric space is a Cauchy sequence.
Definition: A function from a metric space to a metric space is uniformly continuous if for every there exists a such that, for all , implies .
Theorem: A uniformly continuous function carries Cauchy sequences into Cauchy sequences.
Definition: Let be functions from a metric space to a metric space . We say that the sequence converges uniformly to if the following is true: For any there exists (depending on but independent of ) such that for all and all .
Definition: Two metric spaces are said to be homeomorphic if there exists between them a bijection which is continuous in both directions.
Whew! ok, here's what i did.
...oh, wait, i have to go to class. I'll post my "proof" to question 1) later, there's not enough time.
for question 2)
I have no clue, but I have a gut feeling it is true (they wouldn't ask me to prove something that was false, after all)
Ok, so of course that proof was a joke, and a bad one, i know. but i'm not sure what to do