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 $\displaystyle X$ and $\displaystyle Y$ be metric spaces. Assume that $\displaystyle X$ is complete and that there exists $\displaystyle f: X \to Y$ which is bijective and continuous. Suppose that $\displaystyle f^{-1}$ is uniformly continuous. Prove that $\displaystyle Y$ is complete

2) Prove that any open subset of a complete metric space is homeomorphic to a complete metric space. (Hint: If $\displaystyle A$ is the complement of the open set, use the metric

$\displaystyle \left. D_1(x,y) = D(x,y) + \left| \frac 1{D(x,A)} - \frac 1{D(y,A)} \right| \right)$

Here are definitions/theorems i figured would come in handy:

Defintion:A metric space$\displaystyle M$ iscompleteif every Cauchy sequence in$\displaystyle M$ is convergent to a point of $\displaystyle M$.

Theorem:Any convergent sequence in a metric space is a Cauchy sequence.

Definition:A function$\displaystyle f$ from a metric space $\displaystyle X$ to a metric space$\displaystyle Y$ isuniformly continuousif for every$\displaystyle \epsilon > 0$ there exists a$\displaystyle \delta > 0$ such that, for all $\displaystyle x,x' \in X$, $\displaystyle D(x,x') < \delta$ implies$\displaystyle D[f(x),f'(x)] < \epsilon$.

Theorem:A uniformly continuous function carries Cauchy sequences into Cauchy sequences.

Definition:Let$\displaystyle f_i,f$ be functions from a metric space$\displaystyle X$ to a metric space$\displaystyle Y$. We say that the sequence$\displaystyle f_i$converges uniformlyto $\displaystyle f$ if the following is true: For any$\displaystyle \epsilon > 0$ there exists $\displaystyle N$ (depending on $\displaystyle \epsilon$ but independent of $\displaystyle x$) such that $\displaystyle D[f_i(x),f(x)]< \epsilon$ for all $\displaystyle i \ge N$ and all $\displaystyle x \in X$.

Definition:Two metric spaces are said to behomeomorphicif 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)

Proof:

I have no clue, but I have a gut feeling it is true :D (they wouldn't ask me to prove something that was false, after all)

QED

Ok, so of course that proof was a joke, and a bad one, i know. but i'm not sure what to do