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Homework Help: Construct a continuous function in metric space

  1. Apr 12, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex](X,d)[/tex] be a metric space, and let [tex]A,B \subset X[/tex] be disjoint closed subsets.

    1. Construct a continuous function [tex]f : X \to [0,1][/tex] such that [tex]A \subseteq f^{-1}({0})[/tex] and [tex]B \subseteq f^{-1}({1})[/tex]. Hint: use the functions below.

    2. Prove that there are disjoint sets [tex]U,V \subset X[/tex] such that [tex]A \subset U[/tex] and [tex]B \subset V[/tex].

    2. Relevant equations

    [tex]f(x) := d(x,A) = \inf \{ d(x,y) | y \in A \}[/tex] is uniformly continuous on [tex]X[/tex].

    3. The attempt at a solution

    1. I see that for points in [tex]A[/tex] the hint function will always return [tex]0[/tex]. So the preimage of [tex]\{ 0 \}[/tex] is set [tex]A[/tex]. But how can I make the maximum distance from any point to [tex]A[/tex] to [tex]1[/tex] and keep the function continuous? And also how should I make this function return [tex]1[/tex] for all points in [tex]B[/tex]?

    2. I don't understand this (second) part of the question. Is this related to part one? Otherwise it seems obviously true.
  2. jcsd
  3. Apr 12, 2010 #2
    Why don't you try to raise to powers?

    [tex] a^0 =1 [/tex]

    You know how to make the function return 0 for x in B don't you, and you know that x in B will return a finite value d(x,A).
  4. Apr 12, 2010 #3


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    Homework Helper

    some ideas, though they haven't been fully worked...
    - first try drawing the hint function on a 1 d interval
    - then consider the minimum distance between A & B, as they're closed & disjoint, can you say anything about it?
    - how about adding a scale factor & max value to your function

    for the 2nd part i think you can probably use your continuous function, how about considering the preimage under f on open sets.... that said as you say you can probably do it otherwise with just properties of open & closed sets
  5. Apr 12, 2010 #4
    Oh I was thinking

    [tex] d(x,A)^{d(x,B)} [/tex]

    Does this not work
  6. Apr 12, 2010 #5


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    Homework Helper

    nice function & that works in A & B, but what about outside them? might have to clip it to 1? (not that it effects the 2nd part of the question)
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