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Topology question - need help

  1. Jun 26, 2011 #1
    hi all,
    i am studying from croom's introduction to topology book. i came across such a question. and i dont have a clue as to how to start .
    Let X be a metric space with metric d and A a non-empty subset of X. define f:X->IR by :

    f(x): d(x,A), x E X (x is an element of X)
    show that f is continuous.

    also if you can point out a solution book for this book that would be rather nice, considering i am computer scientist studying the topic at home..
    thx.
     
  2. jcsd
  3. Jun 26, 2011 #2

    micromass

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    Hi mbarby! :smile:

    You'll need to prove that

    [tex]d(x,y)<\delta~\Rightarrow~|d(x,A)-d(y,A)|<\varepsilon[/tex]

    Can you first prove that

    [tex]-d(x,y)<d(x,A)-d(y,A)<d(x,y)[/tex]
     
  4. Jun 27, 2011 #3
    to prove that i use the triangular inequality
    d(x,A) <= d(x,y)+d(y,A)
    d(x,A) - d(y,A) <= d(x,y)

    --->
    -d(x,y) <= d(x,A) - d(y,A) <= d(x,y)

    but honestly i couldnt connect it to any kind of a proof :/ ...
     
    Last edited: Jun 27, 2011
  5. Jun 27, 2011 #4

    micromass

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    Doesnt that immediately imply

    [tex]|d(x,A)-d(y,A)|\leq d(x,y)[/tex]

    and this would imply continuity...
     
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