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Criteria for extension to cts fn between metric spaces?

  1. Jun 17, 2010 #1
    Given metric spaces

    [tex](X, d_X), (Y, d_Y)[/tex],

    and subsets

    [tex]\{ x_1, ..., x_n \}, \{ y_1, ..., y_n \}[/tex]

    of X and Y respectively, if I define a function that send [tex]x_i[/tex] to [tex]y_i[/tex], what are the minimal restrictions we need on X and Y in order for there to exist an extension of that map to a continuous function defined on all of X?

    Certainly you'd need some relationship between the connected components of both. So for simplicity, also assume each of the metric spaces are (path) connected. Also assume X and Y are compact.

    If you make any assumptions on Y that will allow you to apply Uryshon's lemma or Tietze extension theorem, then it is not very interesting. I'm curious about the minimal set of restrictions on Y needed to find this extension. (Perhaps I'm missing some elementary theorem that would give my answer?).
     
  2. jcsd
  3. Jun 30, 2010 #2
    If xi,xj lie in the same connected component, so should yi,yj : we can relax the condition of path-connectedness with this restriction.
    Also, I think we need the condition of second countability on Y to construct such a function. I wonder if completeness of Y and X is necessary.
     
  4. Jun 30, 2010 #3
    Thanks for that.

    I'll point out relaxing path-connectedness would fail if you didn't assume Y were compact. E.g., Y = "Topologist's Sine Curve" (see wikipedia), and X = [0, 1]. Then if y_1 = (0,0), y_2 = some other arb pt in Y, and x_1 = 0, x_2 = 1. You'd be unable to extend that to a continuous function, as the compact interval could not be mapped onto the non-compact connected set containing y_1 and y_2.

    I asked a mathematician about my original question, and, well, there response would indicate it's unlikely there's a cut and dried answer to be found.
     
  5. Jun 30, 2010 #4
    In a way, that's right. If X has a property preserved under continuous maps, Y must have the same. The question can be answered only if one specifies what X & Y are.
    I found the question interesting because if extended to countably many points, it would yield a telling result for separable metric spaces.
     
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