Results 1 to 3 of 3

Math Help - completeness

  1. #1
    Member
    Joined
    Dec 2008
    Posts
    154

    completeness

    Let (M_1,d_1) and (M_2,d_2) be metric spaces and let f:M_1 \rightarrow M_2 be a continuous surjective map such that d_1(p,q) \leq d_2(p,q) for evey p,q \in M_1.
    a)If M_1 is complete, is M_2 complete?
    b)If M_2 is complete, is M_1 complete?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1581
    Awards
    1
    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
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Dec 2008
    Posts
    154

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

Similar Math Help Forum Discussions

  1. [SOLVED] Completeness and Consistency
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: August 18th 2011, 03:36 AM
  2. Completeness
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: January 12th 2010, 07:17 AM
  3. NP Completeness
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: December 7th 2009, 01:02 PM
  4. Completeness
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 4th 2009, 11:52 PM
  5. Proving completeness in $L_2(E)$...
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 30th 2008, 08:08 AM

Search Tags


/mathhelpforum @mathhelpforum