Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    lanedance

    User Avatar
    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

    lanedance

    User Avatar
    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)
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook