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Analysis problem

  1. Dec 7, 2006 #1

    kreil

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    Let X be any non-empty set, equipped with the "trural" metric:

    [tex]P_x(x,y)= \{ 0:x=y,1:x \ne y \}[/tex]

    Describe all continuous functions f: X -> R (First describe what it means for a function f to be continuous at a point c in X).

    I'm really quite lost on this, any help is appreciated.

    Josh
     
  2. jcsd
  3. Dec 7, 2006 #2

    AKG

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    What do the open balls in X look like?
     
  4. Dec 7, 2006 #3

    HallsofIvy

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    I'd never seen the term "trural" metric before. I would call that the "discrete" metric. I would also use
    d(x,y) instead of Px(x,y). If x is an argument of the function, what does the subscript x mean?

    As AKG suggested, what do open balls look like? More precisely, what do
    {y| d(x,y)< 1/2} and {y| d(x,y)< 2} look like? What are the open sets in this metric?

    I presume your definition of "continuous function" is one for which f-1(C) is open whenever C is open.
     
  5. Dec 7, 2006 #4
    yeah. trural threw me too. if it's the discrete (as it appears, though the notation is odd) all sets are open and the question is easy
     
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