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Math Help - The discrete Metric- Completeness and Compactness

  1. #1
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    The discrete Metric- Completeness and Compactness

    Hello, I'm just working through this problem but I am a little shaky with some of the definitions

    Let M be a set and let d be the discrete metric on
    M.
    1. Show that (M, d) is complete.
    2. Describe all the compact subsets of M.

    Since (M,d) is equipped with the discrete metric all sequences inside that metric converge to some pt that repeats. How is this cauchy?
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  2. #2
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by Scopur View Post
    Hello, I'm just working through this problem but I am a little shaky with some of the definitions

    Let M be a set and let d be the discrete metric on
    M.
    1. Show that (M, d) is complete.
    2. Describe all the compact subsets of M.

    Since (M,d) is equipped with the discrete metric all sequences inside that metric converge to some pt that repeats. How is this cauchy?
    hm... every converging sequence is Cauchy sequence


    Th : "every compact metric space is complete .... "

    if X is compact metric space and let  (x_n) be Cauchy sequence in X. because of compactness there is sub sequence  x_{nk} of our sequence.. and sub sequence is converging  x_{nk} \to x_0 \in X    \;\;   (k \to \infty ) now we have

     d(x_n,x_0) <= d(x_n, x_{nk} ) + d ( x_{nk} , x_0)

    now first on the right side can be made very small because it is Cauchy , and second to because of convergence of the sub sequence so we have

     d(x_n,x_0) \to 0 \;\; (n\to \infty)

    meaning sequence x_n is converging and space X is complete
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  3. #3
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    for part b) use the fact that given any open cover of a compact set, there exists a finite subcover. Open balls of radius one about x, are denoted \ B_1 (x) and \forall  x  \in  M , \ B_1 (x) = \{x\}
    so only finite subsets are compact.
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