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Thread: completeness

  1. #1
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    completeness

    Let $\displaystyle (M_1,d_1)$ and $\displaystyle (M_2,d_2)$ be metric spaces and let $\displaystyle f:M_1 \rightarrow M_2$ be a continuous surjective map such that $\displaystyle d_1(p,q) \leq d_2(p,q)$ for evey $\displaystyle p,q \in M_1$.
    a)If $\displaystyle M_1 $ is complete, is $\displaystyle M_2$ complete?
    b)If $\displaystyle M_2$ is complete, is $\displaystyle M_1 $ complete?
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  2. #2
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    Kat-M
    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
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  3. #3
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    questions

    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 $\displaystyle M_2$ is complete.
    Let $\displaystyle {x_n} $ be Caushy in $\displaystyle M_1 $.And using the fact that $\displaystyle f$ is const, for $\displaystyle \epsilon \geq 0$ there exists $\displaystyle N$ such that $\displaystyle n,m \geq N$ implies $\displaystyle d_1(x_n,x_m)< \delta$ such that $\displaystyle d_2(f(x_n),f(x_m)) < \epsilon$. so $\displaystyle \{f(x_n)\}$ is also Caushy in $\displaystyle M_2$.
    Since $\displaystyle M_2$ is complete, $\displaystyle f(x_n) \rightarrow y \in M_2$. Since f is surjective, there exists $\displaystyle x \in M_1$ such that $\displaystyle f(x)=y$.
    Since $\displaystyle \{f(x_n)\}$ converges to $\displaystyle y$, for $\displaystyle \epsilon > 0$, there exists $\displaystyle M$ such that $\displaystyle n \geq M$ implies $\displaystyle d_2(f(x_n),y) < \epsilon$. Since $\displaystyle d_1 \leq d_2$, $\displaystyle d(x_n, x) < \epsilon$. So $\displaystyle \{x_n\}$ converges to $\displaystyle x$. Thus $\displaystyle (M_1, d_1)$ is complete.

    I am stuck in part 1 so if you can help me with this, i am really greatful. Thank you.
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