Hamming metric

  1. 1. The problem statement, all variables and given/known data

    I'm stuck on how to start this. The Hammin metric is define:
    http://s1038.photobucket.com/albums/a467/kanye_brown/?action=view&current=hamming_metric.jpg

    and I'm asked to:
    http://i1038.photobucket.com/albums/a467/kanye_brown/analysis_1.jpg?t=1306280360

    a) prove the set U(d1,...,dp) is an open subset of X.


    b) Prove that U is a basis of open sets for (X, d).

    c) Say whether the statement is true or false.

    Consider the following statement: “For every p 2 N and every d1, . . . , dp ϵ {0, 1},
    the set U(d1,...,dp) is a closed subset of X.”
    Is the statement true? Justify your answer (with a proof or counterexample).

    Any ideas?



    2. Relevant equations



    3. The attempt at a solution

    I'm not sure where to start. I know (X,d) is a metric space but that's about it so far.
     
  2. jcsd
  3. micromass

    micromass 18,533
    Staff Emeritus
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    You can start by describing the balls in this space. Let's first answer an easy question:

    given x=(0,0,0,...), can you find me all sequences y such d(x,y)<1/2?
     
  4. What I've figured out is the set is almost an open ball around the sequence d1,d2,d3...d_p of
    radius (1/2)^p.

    I think there may be a special case where everything is the same except for x_p
     
  5. micromass

    micromass 18,533
    Staff Emeritus
    Science Advisor

    Well, maybe your open set is the union of two balls?
     
  6. I'm not sure I fully follow -- could you elaborate?
     
  7. micromass

    micromass 18,533
    Staff Emeritus
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    Well, can you first tell me what the ball around (0,0,0,...) with radius 1/2 looks like?
     
  8. not sure...just a point?
     
  9. micromass

    micromass 18,533
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    Science Advisor

    No, you'll need to find all [tex](x_1,x_2,x_3,...)[/tex] such that

    [tex]\sum_{i=1}^{+\infty}{\frac{x_i}{2^i}}<\frac{1}{2}[/tex]

    For example, (0,1,0,0,...) is such a point, (0,1,0,1,0,1,0,1,...) too...
     
  10. Still a bit stuck...
     
  11. micromass

    micromass 18,533
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    OK, let's investigate the situation. For which vectors (x1,x2,x3,...) does

    [tex]\sum_{k=0}^{+\infty}{\frac{x_k}{2^k}}=1[/tex]?
     
  12. I guess ones whose infinite sum is one, ie. Things of the form 1/2 + 1/4 + ....

    Not sure how to get that back into the corresponding x_k
     
  13. micromass

    micromass 18,533
    Staff Emeritus
    Science Advisor

    Well, once you figured that out, we can take the next step! :smile:
     
  14. Thanks -- what about b) ...

    I don't really understand the concept of basis of open sets
     
  15. micromass

    micromass 18,533
    Staff Emeritus
    Science Advisor

    To solve (b), you will also need to know what the open balls look like. So you'll need to figure that out first.

    A basis is just a collection of open sets, such that every open set can be shrunk to a basis open set. More formally: for every x and for every set G around x, we can find a basis element B such that [itex]x\in B\subeteq G[/itex].

    The trick is that we don't need to show this for open sets in general (this would be too hard), but it suffices to look at balls. But of course, you'll need to know the balls for that...
     
  16. ...care to give me a hint on what the balls look like?

    I must admit, I find the notation confusing
     
  17. micromass

    micromass 18,533
    Staff Emeritus
    Science Advisor

    Well, I want to figure out together what the balls look like. But for that, you'll first need to answer my post 10, since this is the first to find what the balls look like...
     
  18. Is it points with the only the even entries allowed to equal to 1 and the odd entries equal to 0?

    so the 2nd, 4th, 6th, etc could be 1 or 0
    but 1st, 3rd, 5th, etc can only be 0?
     
  19. micromass

    micromass 18,533
    Staff Emeritus
    Science Advisor

    No, because (0,1,0,1,0,1,0,1,...) would correspond to

    [tex]\sum_{k=1}^{+\infty}{\frac{1}{2^{2k}}=\sum_{k=1}^{+\infty}{\frac{1}{4^k}}=\frac{1}{3}[/tex]

    and this is not 1.
     
  20. Hmm...I'm stuck
     
  21. any suggestions on materials I can read? I'm not sure I can visualize this
     
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